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TriForce

Team Members

1. David Ferri, Sudoku Solver
2. Vincent Terpstra, Julia Sets
3. Raymond Kiguru, EasyBMP

Progress

Assignment 1: Sudoku Solver

Sudoku Solver Profiling

Rather than try to continuously increase the difficulty of a 9x9 sudoku, I decided to modify the program I found to handle larger and large sudokus, increasing the size of the matrices that make up the sudoku (starting with a 9x9 sudoku, which is 9 3x3 matrices, then 16x16 which is 16 4x4 matrices, and finally 25x25 which is 25 5x5 matrices) without changing the logic of the program (only constants), so larger sudokus are solved the same way as a normal one.

Source code from: https://www.geeksforgeeks.org/sudoku-backtracking-7/

Original Code:

```// A Backtracking program  in C++ to solve Sudoku problem
#include <stdio.h>

// UNASSIGNED is used for empty cells in sudoku grid
#define UNASSIGNED 0

// N is used for the size of Sudoku grid. Size will be NxN
#define N 9

// This function finds an entry in grid that is still unassigned
bool FindUnassignedLocation(int grid[N][N], int &row, int &col);

// Checks whether it will be legal to assign num to the given row, col
bool isSafe(int grid[N][N], int row, int col, int num);

/* Takes a partially filled-in grid and attempts to assign values to
all unassigned locations in such a way to meet the requirements
for Sudoku solution (non-duplication across rows, columns, and boxes) */
bool SolveSudoku(int grid[N][N])
{
int row, col;

// If there is no unassigned location, we are done
if (!FindUnassignedLocation(grid, row, col))
return true; // success!

// consider digits 1 to 9
for (int num = 1; num <= 9; num++)
{
// if looks promising
if (isSafe(grid, row, col, num))
{
// make tentative assignment
grid[row][col] = num;

// return, if success, yay!
if (SolveSudoku(grid))
return true;

// failure, unmake & try again
grid[row][col] = UNASSIGNED;
}
}
return false; // this triggers backtracking
}

/* Searches the grid to find an entry that is still unassigned. If
found, the reference parameters row, col will be set the location
that is unassigned, and true is returned. If no unassigned entries
remain, false is returned. */
bool FindUnassignedLocation(int grid[N][N], int &row, int &col)
{
for (row = 0; row < N; row++)
for (col = 0; col < N; col++)
if (grid[row][col] == UNASSIGNED)
return true;
return false;
}

/* Returns a boolean which indicates whether an assigned entry
in the specified row matches the given number. */
bool UsedInRow(int grid[N][N], int row, int num)
{
for (int col = 0; col < N; col++)
if (grid[row][col] == num)
return true;
return false;
}

/* Returns a boolean which indicates whether an assigned entry
in the specified column matches the given number. */
bool UsedInCol(int grid[N][N], int col, int num)
{
for (int row = 0; row < N; row++)
if (grid[row][col] == num)
return true;
return false;
}

/* Returns a boolean which indicates whether an assigned entry
within the specified 3x3 box matches the given number. */
bool UsedInBox(int grid[N][N], int boxStartRow, int boxStartCol, int num)
{
for (int row = 0; row < 3; row++)
for (int col = 0; col < 3; col++)
if (grid[row+boxStartRow][col+boxStartCol] == num)
return true;
return false;
}

/* Returns a boolean which indicates whether it will be legal to assign
num to the given row,col location. */
bool isSafe(int grid[N][N], int row, int col, int num)
{
/* Check if 'num' is not already placed in current row,
current column and current 3x3 box */
return !UsedInRow(grid, row, num)&&!UsedInCol(grid, col, num)&&!UsedInBox(grid, row - row%3 , col - col%3, num)&&grid[row][col]==UNASSIGNED;
}

/* A utility function to print grid  */
void printGrid(int grid[N][N])
{
for (int row = 0; row < N; row++)
{
for (int col = 0; col < N; col++)
printf("%2d", grid[row][col]);
printf("\n");
}
}

/* Driver Program to test above functions */
int main()
{
// 0 means unassigned cells
int grid[N][N] = {{3, 0, 6, 5, 0, 8, 4, 0, 0},
{5, 2, 0, 0, 0, 0, 0, 0, 0},
{0, 8, 7, 0, 0, 0, 0, 3, 1},
{0, 0, 3, 0, 1, 0, 0, 8, 0},
{9, 0, 0, 8, 6, 3, 0, 0, 5},
{0, 5, 0, 0, 9, 0, 6, 0, 0},
{1, 3, 0, 0, 0, 0, 2, 5, 0},
{0, 0, 0, 0, 0, 0, 0, 7, 4},
{0, 0, 5, 2, 0, 6, 3, 0, 0}};
if (SolveSudoku(grid) == true)
printGrid(grid);
else
printf("No solution exists");

return 0;
}
```

Modified Version for 16x16 Puzzle:

```// A Backtracking program  in C++ to solve Sudoku problem
#include <stdio.h>

// UNASSIGNED is used for empty cells in sudoku grid
#define UNASSIGNED 0

// N is used for the size of Sudoku grid. Size will be NxN
#define N 16

// This function finds an entry in grid that is still unassigned
bool FindUnassignedLocation(int grid[N][N], int &row, int &col);

// Checks whether it will be legal to assign num to the given row, col
bool isSafe(int grid[N][N], int row, int col, int num);

/* Takes a partially filled-in grid and attempts to assign values to
all unassigned locations in such a way to meet the requirements
for Sudoku solution (non-duplication across rows, columns, and boxes) */
bool SolveSudoku(int grid[N][N])
{
int row, col;

// If there is no unassigned location, we are done
if (!FindUnassignedLocation(grid, row, col))
return true; // success!

// consider digits 1 to 16
for (int num = 1; num <= 16; num++)
{
// if looks promising
if (isSafe(grid, row, col, num))
{
// make tentative assignment
grid[row][col] = num;

// return, if success, yay!
if (SolveSudoku(grid))
return true;

// failure, unmake & try again
grid[row][col] = UNASSIGNED;
}
}
return false; // this triggers backtracking
}

/* Searches the grid to find an entry that is still unassigned. If
found, the reference parameters row, col will be set the location
that is unassigned, and true is returned. If no unassigned entries
remain, false is returned. */
bool FindUnassignedLocation(int grid[N][N], int &row, int &col)
{
for (row = 0; row < N; row++)
for (col = 0; col < N; col++)
if (grid[row][col] == UNASSIGNED)
return true;
return false;
}

/* Returns a boolean which indicates whether an assigned entry
in the specified row matches the given number. */
bool UsedInRow(int grid[N][N], int row, int num)
{
for (int col = 0; col < N; col++)
if (grid[row][col] == num)
return true;
return false;
}

/* Returns a boolean which indicates whether an assigned entry
in the specified column matches the given number. */
bool UsedInCol(int grid[N][N], int col, int num)
{
for (int row = 0; row < N; row++)
if (grid[row][col] == num)
return true;
return false;
}

/* Returns a boolean which indicates whether an assigned entry
within the specified 4x4 box matches the given number. */
bool UsedInBox(int grid[N][N], int boxStartRow, int boxStartCol, int num)
{
for (int row = 0; row < 4; row++)
for (int col = 0; col < 4; col++)
if (grid[row+boxStartRow][col+boxStartCol] == num)
return true;
return false;
}

/* Returns a boolean which indicates whether it will be legal to assign
num to the given row,col location. */
bool isSafe(int grid[N][N], int row, int col, int num)
{
/* Check if 'num' is not already placed in current row,
current column and current 4x4 box */
return !UsedInRow(grid, row, num)&&!UsedInCol(grid, col, num)&&!UsedInBox(grid, row - row%4 , col - col%4, num)&&grid[row][col]==UNASSIGNED;
}

/* A utility function to print grid  */
void printGrid(int grid[N][N])
{
for (int row = 0; row < N; row++)
{
for (int col = 0; col < N; col++)
printf("%2d", grid[row][col]);
printf("\n");
}
}

/* Driver Program to test above functions */
int main()
{
//https://puzzlemadness.co.uk/16by16giantsudoku/
// 0 means unassigned cells
int grid[N][N] = {{0,  8,   0,  0,  0,  0,  0,  3,  0,  0,  0, 10,  9,  7, 11, 0},
{0,  9,  15, 13,  0, 10,  0,  0,  2,  6,  8, 16,  0,  0,  0, 0},
{0,  0,  16,  0, 15,  0,  8,  0,  9,  0,  0,  0,  6,  0,  2, 0},
{1,  0,   2,  0,  9, 11,  4,  6, 15,  3,  5,  7,  0,  0, 12, 0},
{16, 6,   4,  0,  5,  2,  0,  0,  1,  0,  0,  0, 11,  0,  0, 12},
{5,  11,  0,  0,  0,  3,  0, 15,  0, 16,  0, 13,  0,  1,  0, 8},
{0,  0,   3,  0,  0,  6, 11, 14,  0,  5,  7,  0,  0,  9,  0, 0},
{0,  0,   0, 14,  8,  0, 10,  0,  0, 11, 12,  0,  0,  0,  0, 0},
{0,  7,  13,  0,  0,  0,  0, 12,  0,  8,  9,  0,  0,  0,  3, 0},
{0,  0,  11,  9,  0,  7,  0,  0,  0,  0,  0, 12,  0,  8, 16, 5},
{0,  0,  10,  0, 11, 13,  0,  0,  0,  0,  0,  3, 12,  0,  6, 0},
{0,  5,   0,  0, 10, 15,  0,  1,  7,  2,  0,  0, 14, 11,  0, 0},
{0,  0,   5,  0,  0, 12, 14,  0,  0, 10,  0,  0, 15,  0,  0, 4},
{9,  0,  14,  6,  0,  0,  1,  0, 16,  0,  2,  0,  3,  0, 13, 0},
{8,  13,  0,  4,  0,  0,  0,  0, 12,  7,  3,  0,  0,  6,  0, 0},
{0,  16, 12,  0,  0,  5,  0,  9,  0, 13, 14,  4,  1,  0,  0, 0}};
if (SolveSudoku(grid) == true)
printGrid(grid);
else
printf("No solution exists");

return 0;
}
```

Modified Version for 25x25 puzzle:

```// A Backtracking program  in C++ to solve Sudoku problem
#include <stdio.h>

// UNASSIGNED is used for empty cells in sudoku grid
#define UNASSIGNED 0

// N is used for the size of Sudoku grid. Size will be NxN
#define N 25

// This function finds an entry in grid that is still unassigned
bool FindUnassignedLocation(int grid[N][N], int &row, int &col);

// Checks whether it will be legal to assign num to the given row, col
bool isSafe(int grid[N][N], int row, int col, int num);

/* Takes a partially filled-in grid and attempts to assign values to
all unassigned locations in such a way to meet the requirements
for Sudoku solution (non-duplication across rows, columns, and boxes) */
bool SolveSudoku(int grid[N][N])
{
int row, col;

// If there is no unassigned location, we are done
if (!FindUnassignedLocation(grid, row, col))
return true; // success!

// consider digits 1 to 25
for (int num = 1; num <= 25; num++)
{
// if looks promising
if (isSafe(grid, row, col, num))
{
// make tentative assignment
grid[row][col] = num;

// return, if success, yay!
if (SolveSudoku(grid))
return true;

// failure, unmake & try again
grid[row][col] = UNASSIGNED;
}
}
return false; // this triggers backtracking
}

/* Searches the grid to find an entry that is still unassigned. If
found, the reference parameters row, col will be set the location
that is unassigned, and true is returned. If no unassigned entries
remain, false is returned. */
bool FindUnassignedLocation(int grid[N][N], int &row, int &col)
{
for (row = 0; row < N; row++)
for (col = 0; col < N; col++)
if (grid[row][col] == UNASSIGNED)
return true;
return false;
}

/* Returns a boolean which indicates whether an assigned entry
in the specified row matches the given number. */
bool UsedInRow(int grid[N][N], int row, int num)
{
for (int col = 0; col < N; col++)
if (grid[row][col] == num)
return true;
return false;
}

/* Returns a boolean which indicates whether an assigned entry
in the specified column matches the given number. */
bool UsedInCol(int grid[N][N], int col, int num)
{
for (int row = 0; row < N; row++)
if (grid[row][col] == num)
return true;
return false;
}

/* Returns a boolean which indicates whether an assigned entry
within the specified 5x5 box matches the given number. */
bool UsedInBox(int grid[N][N], int boxStartRow, int boxStartCol, int num)
{
for (int row = 0; row < 5; row++)
for (int col = 0; col < 5; col++)
if (grid[row+boxStartRow][col+boxStartCol] == num)
return true;
return false;
}

/* Returns a boolean which indicates whether it will be legal to assign
num to the given row,col location. */
bool isSafe(int grid[N][N], int row, int col, int num)
{
/* Check if 'num' is not already placed in current row,
current column and current 5x5 box */
return !UsedInRow(grid, row, num)&&!UsedInCol(grid, col, num)&&!UsedInBox(grid, row - row%5 , col - col%5, num)&&grid[row][col]==UNASSIGNED;
}

/* A utility function to print grid  */
void printGrid(int grid[N][N])
{
for (int row = 0; row < N; row++)
{
for (int col = 0; col < N; col++)
printf("%2d", grid[row][col]);
printf("\n");
}
}

/* Driver Program to test above functions */
int main()
{
//http://www.sudoku-download.net/sudoku_25x25.php
// 0 means unassigned cells
int grid[N][N] = {{1,  0,   4,  0, 25,  0, 19,  0,  0,  10,  21, 8,  0,  14, 0,  6,  12,   9,  0,  0,  0,  0,  0,  0,  5},
{5,  0,  19, 23, 24,  0, 22,  12,  0,  0,  16, 6,  0,  20,  0,  18,  0,   25,  14,  13,  10, 11,  0,  1,  15},
{0,  0,   0,  0,  0,  0,  21,  5,  0,  20,  11,  10,  0,  1,  0,  4,  8,   24,  23,  15,  18,  0,  16,  22,  19},
{0,  7,  21,  8, 18,  0,  0,  0, 11,  0,  5,  0,  0,  24, 0,  0,  0,   17,  22,  1,  9,  6,  25,  0,  0},
{0, 13,  15,  0, 22, 14,  0,  18,  0,  16,  0,  0, 0,  4,  0,  0,  0,   19,  0,  0,  0,  24,  20,  21,  17},
{12, 0,  11,  0,  6,  0,  0, 0,  0, 15,  0, 0,  0,  0,  21,  25,  19,   0,  4,  0,  22,  14,  0,  20,  0},
{8,  0,   0, 21,  0, 16, 0, 0,  0,  2,  0,  3,  0,  0,  0,  0,  17,   23,  18,  22,  0,  0,  0,  24,  6},
{4,  0,  14, 18,  7,  9, 0,  22,  21, 19, 0,  0,  0,  2,  0,  5,  0,   0,  0,  6,  16,  15,  0,  11,  12},
{22, 0,  24,  0, 23,  0,  0, 11,  0,  7,  0,  0,  4,  0,  14,  0,  2,   12,  0,  8,  5,  19,  0,  25,  9},
{20, 0,   0,  0,  5,  0,  0,  0,  0,  17,  9, 0,  12,  18, 0,  1,  0,   0,  7,  24,  0,  0,  0,  13,  4},
{13, 0,   0,  5,  0,  2,  23,  14,  4,  18,  22,  0, 17,  0,  0,  20,  0,   1,  9,  21,  12,  0,  0,  8,  11},
{14, 23,  0, 24,  0,  0,  0,  0,  0,  0,  0,  0, 20, 25,  0,  3,  4,   13,  0,  11,  21,  9,  5,  18,  22},
{7,  0,   0, 11, 17, 20, 24,  0,  0, 0,  3,  4, 1,  12,  0,  0,  6,   14,  0,  5,  25,  13,  0,  0,  0},
{0,  0,  16,  9,  0, 17,  11,  7, 10,  25,  0,  0,  0,  13, 6,  0,  0,   18,  0,  0,  19,  4,  0,  0,  20},
{6,  15,  0, 19,  4, 13,  0,  0, 5,  0,  18,  11,  0,  0,  9,  8,  22,   16,  25,  10,  7,  0,  0,  0,  0},
{0,  0,   0,  2,  0,  0, 10, 19,  3,  0,  1,  0,  22,  9,  4,  11,  15,   0,  20,  0,  0,  8,  23,  0,  25},
{0,  24,  8, 13,  1,  0, 0,  4,  20, 0, 17,  14,  0,  0,  18,  0,  16,   22,  5,  0,  11,  0,  10,  0,  0},
{23, 10,  0,  0,  0,  0,  0, 0,  18,  0,  6,  0,  16,  0,  0,  17,  1,   0,  13,  0,  0,  3,  19,  12,  0},
{25,  5,  0, 14, 11,  0,  17,  0,  8,  24,  13, 0,  19,  23, 15,  9,  0,   0,  12,  0,  20,  0,  22,  0,  7},
{0,   0, 17,  4,  0, 22,  15,  0,  23,  11,  12,  25, 0,  0,  0,  0,  18,   8,  0,  7,  0,  0,  14,  0,  13},
{19,  6, 23, 22,  8,  0,  0,  1,  25,  4,  14,  2, 0, 3,  7,  13,  10,   11,  16,  0,  0,  0,  0,  0,  0},
{0,   4,  0, 17,  0,  3, 0,  24,  0, 8,  20,  23, 11,  10,  25,  22,  0,   0,  0,  12,  13,  2,  18,  6,  0},
{0,   0,  7, 16,  0,  0,  6,  17, 2,  21,  0,  18,  0,  0, 0,  19,  0,   0,  8,  0,  0,  0,  0,  4,  0},
{18,  9, 25,  1,  2, 11,  0,  0, 13,  22,  4,  0,  21,  0,  5,  0,  23,   7,  0,  0,  15,  0,  3,  0,  8},
{0,  21, 10,  0,  0, 12,  0,  20,  16, 0, 19,  0,  0,  0,  0,  15,  14,   4,  2,  18,  23,  25,  11,  7,  0}};
if (SolveSudoku(grid) == true)
printGrid(grid);
else
printf("No solution exists");

return 0;
}
```

Obtaining flat profiles and call graphs on matrix environment:

```\$ g++ sudokuC.cpp  -std=c++0x -o Sudoku
\$ ./Sudoku
3 1 6 5 7 8 4 9 2
5 2 9 1 3 4 7 6 8
4 8 7 6 2 9 5 3 1
2 6 3 4 1 5 9 8 7
9 7 4 8 6 3 1 2 5
8 5 1 7 9 2 6 4 3
1 3 8 9 4 7 2 5 6
6 9 2 3 5 1 8 7 4
7 4 5 2 8 6 3 1 9
\$ gprof -p -b ./Sudoku gmon.out > 9x9.flt
\$ gprof -q -b ./Sudoku gmon.out > 9x9.clg
```
```\$  g++ sudokuC16.cpp  -std=c++0x -pg -o Sudoku16
\$ ./Sudoku16
12 8 6 516 1 2 31314 410 9 71115
11 915131210 7 5 2 6 816 414 1 3
4 316 71514 813 91211 1 6 5 210
114 210 911 4 615 3 5 7 8131216
16 6 415 5 213 7 1 910 811 31412
511 9 2 4 312151416 613 7 110 8
1012 3 8 1 61114 4 5 7 216 91513
13 1 714 8 91016 3111215 2 4 5 6
2 71316 6 4 51211 8 9141015 3 1
6 411 914 7 3 210 1151213 816 5
141510 11113 9 8 5 416 312 2 6 7
3 5 812101516 1 7 213 61411 4 9
7 2 5 313121411 610 1 91516 8 4
91014 6 7 8 1 41615 2 5 3121311
813 1 4 216151012 7 311 5 6 914
15161211 3 5 6 9 81314 4 110 7 2
\$ gprof -p -b ./Sudoku16 gmon.out > 16x16.flt
\$ gprof -q -b ./Sudoku16 gmon.out > 16x16.clg
```
```\$  g++ sudokuC25.cpp  -std=c++0x -pg -o Sudoku25
\$ ./Sudoku25
111 42025241915171021 8181422 612 9 316 2 71323 5
5 2192324 82212 9 316 6 7201718212514131011 4 115
1714 9 6 32521 5 7201110 2 113 4 82423151812162219
16 721 818 4 2131123 51915241210201722 1 9 62514 3
101315122214 118 61623 925 4 3 7 51911 2 824202117
12 11110 6 513232415 716 817212519 3 4 92214 22018
8191321 916 42512 215 3 511201417231822 110 724 6
4171418 7 9 322211925 124 223 5132010 61615 81112
22 3241523182011 1 71013 4 61416 21221 8 5191725 9
2016 225 510 8 61417 922121819 11115 724 3232113 4
1325 3 510 22314 418221517192420 7 1 9211216 6 811
1423 124121916 815 6 2 7202510 3 413171121 9 51822
7 818111720242122 9 3 4 11216 2 61419 52513151023
22216 9211711 71025 8 51413 6122418152319 4 1 320
6152019 41312 3 5 118112321 9 822162510 71724 214
211812 216 71019 313 12422 9 41115 6201417 823 525
924 813 1 625 420121714 3 718231622 51911211015 2
231022 71521 5 91814 62016 81117 1 21325 4 3191224
25 5 61411 117 2 8241321192315 9 31012 420182216 7
32017 4192215162311122510 5 22118 824 7 6 114 913
19 62322 81518 125 414 2 9 3 7131011162024 5121721
15 4 51714 3 72419 8202311102522 921 11213 218 616
1112 7162023 617 22124181315 11925 5 8 31422 9 410
18 925 1 21114101322 4122116 52423 7 6171520 319 8
242110 31312 92016 51917 622 81514 4 218232511 7 1
\$ gprof -p -b ./Sudoku25 gmon.out > 25x25.flt
\$ gprof -q -b ./Sudoku25 gmon.out > 25x25.clg
```

For 9x9 Sudoku Puzzle (3x3 squares)

```Flat profile:
Each sample counts as 0.01 seconds.
no time accumulated

%   cumulative   self              self     total
time   seconds   seconds    calls  Ts/call  Ts/call  name
0.00      0.00     0.00     6732     0.00     0.00  isSafe(int (*) [9], int, int, int)
0.00      0.00     0.00     6732     0.00     0.00  UsedInRow(int (*) [9], int, int)
0.00      0.00     0.00     2185     0.00     0.00  UsedInCol(int (*) [9], int, int)
0.00      0.00     0.00     1078     0.00     0.00  UsedInBox(int (*) [9], int, int, int)
0.00      0.00     0.00      770     0.00     0.00  FindUnassignedLocation(int (*) [9], int&, int&)
0.00      0.00     0.00        1     0.00     0.00  SolveSudoku(int (*) [9])
0.00      0.00     0.00        1     0.00     0.00  printGrid(int (*) [9])
```
```Call graph

granularity: each sample hit covers 2 byte(s) no time propagated

index % time    self  children    called     name
0.00    0.00    6732/6732        SolveSudoku(int (*) [9]) [13]
[8]      0.0    0.00    0.00    6732         isSafe(int (*) [9], int, int, int) [8]
0.00    0.00    6732/6732        UsedInRow(int (*) [9], int, int) [9]
0.00    0.00    2185/2185        UsedInCol(int (*) [9], int, int) [10]
0.00    0.00    1078/1078        UsedInBox(int (*) [9], int, int, int) [11]
-----------------------------------------------
0.00    0.00    6732/6732        isSafe(int (*) [9], int, int, int) [8]
[9]      0.0    0.00    0.00    6732         UsedInRow(int (*) [9], int, int) [9]
-----------------------------------------------
0.00    0.00    2185/2185        isSafe(int (*) [9], int, int, int) [8]
[10]     0.0    0.00    0.00    2185         UsedInCol(int (*) [9], int, int) [10]
-----------------------------------------------
0.00    0.00    1078/1078        isSafe(int (*) [9], int, int, int) [8]
[11]     0.0    0.00    0.00    1078         UsedInBox(int (*) [9], int, int, int) [11]
-----------------------------------------------
0.00    0.00     770/770         SolveSudoku(int (*) [9]) [13]
[12]     0.0    0.00    0.00     770         FindUnassignedLocation(int (*) [9], int&, int&) [12]
-----------------------------------------------
769             SolveSudoku(int (*) [9]) [13]
0.00    0.00       1/1           main [6]
[13]     0.0    0.00    0.00       1+769     SolveSudoku(int (*) [9]) [13]
0.00    0.00    6732/6732        isSafe(int (*) [9], int, int, int) [8]
0.00    0.00     770/770         FindUnassignedLocation(int (*) [9], int&, int&) [12]
769             SolveSudoku(int (*) [9]) [13]
-----------------------------------------------
0.00    0.00       1/1           main [6]
[14]     0.0    0.00    0.00       1         printGrid(int (*) [9]) [14]
-----------------------------------------------
Index by function name
[13] SolveSudoku(int (*) [9]) [11] UsedInBox(int (*) [9], int, int, int) [14] printGrid(int (*) [9])
[12] FindUnassignedLocation(int (*) [9], int&, int&) [10] UsedInCol(int (*) [9], int, int)
[8] isSafe(int (*) [9], int, int, int) [9] UsedInRow(int (*) [9], int, int)
```

For 16x16 Sudoku Puzzle (4x4 squares) Puzzle from: [1]

```Flat profile:
Each sample counts as 0.01 seconds.
%   cumulative   self              self     total
time   seconds   seconds    calls   s/call   s/call  name
39.04     15.00    15.00 28071636     0.00     0.00  FindUnassignedLocation(int (*) [16], int&, int&)
36.19     28.90    13.90 449145092     0.00     0.00  UsedInRow(int (*) [16], int, int)
10.60     32.97     4.07 120354547     0.00     0.00  UsedInCol(int (*) [16], int, int)
4.97     34.88     1.91 41212484     0.00     0.00  UsedInBox(int (*) [16], int, int, int)
4.59     36.65     1.76        1     1.76    38.39  SolveSudoku(int (*) [16])
4.55     38.39     1.75 449145092     0.00     0.00  isSafe(int (*) [16], int, int, int)
0.01     38.40     0.01                             frame_dummy
0.00     38.40     0.00        1     0.00     0.00  printGrid(int (*) [16])
```
```Call graph

granularity: each sample hit covers 2 byte(s) for 0.03% of 36.85 seconds

index % time    self  children    called     name
<spontaneous>
[1]    100.0    0.00   36.85                 main [1]
1.93   34.93       1/1           SolveSudoku(int (*) [16]) [2]
0.00    0.00       1/1           printGrid(int (*) [16]) [14]
-----------------------------------------------
28071635             SolveSudoku(int (*) [16]) [2]
1.93   34.93       1/1           main [1]
[2]    100.0    1.93   34.93       1+28071635 SolveSudoku(int (*) [16]) [2]
1.69   19.09 449145092/449145092     isSafe(int (*) [16], int, int, int) [3]
14.14    0.00 28071636/28071636     FindUnassignedLocation(int (*) [16], int&, int&) [4]
28071635             SolveSudoku(int (*) [16]) [2]
-----------------------------------------------
1.69   19.09 449145092/449145092     SolveSudoku(int (*) [16]) [2]
[3]     56.4    1.69   19.09 449145092         isSafe(int (*) [16], int, int, int) [3]
13.58    0.00 449145092/449145092     UsedInRow(int (*) [16], int, int) [5]
3.54    0.00 120354547/120354547     UsedInCol(int (*) [16], int, int) [6]
1.98    0.00 41212484/41212484     UsedInBox(int (*) [16], int, int, int) [7]
-----------------------------------------------
14.14    0.00 28071636/28071636     SolveSudoku(int (*) [16]) [2]
[4]     38.4   14.14    0.00 28071636         FindUnassignedLocation(int (*) [16], int&, int&) [4]
-----------------------------------------------
13.58    0.00 449145092/449145092     isSafe(int (*) [16], int, int, int) [3]
[5]     36.8   13.58    0.00 449145092         UsedInRow(int (*) [16], int, int) [5]
-----------------------------------------------
3.54    0.00 120354547/120354547     isSafe(int (*) [16], int, int, int) [3]
[6]      9.6    3.54    0.00 120354547         UsedInCol(int (*) [16], int, int) [6]
-----------------------------------------------
1.98    0.00 41212484/41212484     isSafe(int (*) [16], int, int, int) [3]
[7]      5.4    1.98    0.00 41212484         UsedInBox(int (*) [16], int, int, int) [7]
-----------------------------------------------
0.00    0.00       1/1           main [1]
[14]     0.0    0.00    0.00       1         printGrid(int (*) [16]) [14]
-----------------------------------------------
Index by function name
[2] SolveSudoku(int (*) [16]) [7] UsedInBox(int (*) [16], int, int, int) [14] printGrid(int (*) [16])
[4] FindUnassignedLocation(int (*) [16], int&, int&) [6] UsedInCol(int (*) [16], int, int)
[3] isSafe(int (*) [16], int, int, int) [5] UsedInRow(int (*) [16], int, int)
```

For 25x25 Sudoku Puzzle (5x5 squares) Puzzle from: http://www.sudoku-download.net/sudoku_25x25.php

```Flat profile:
Each sample counts as 0.01 seconds.
%   cumulative   self              self     total
time   seconds   seconds    calls  Ks/call  Ks/call  name
48.76   1052.18  1052.18 425478951     0.00     0.00  UsedInRow(int (*) [25], int, int)
25.24   1596.81   544.63 876012758     0.00     0.00  FindUnassignedLocation(int (*) [25], int&, int&)
12.48   1866.03   269.21 590817023     0.00     0.00  UsedInCol(int (*) [25], int, int)
4.83   1970.24   104.21 425478951     0.00     0.00  isSafe(int (*) [25], int, int, int)
4.79   2073.51   103.27        1     0.10     2.17  SolveSudoku(int (*) [25])
4.35   2167.39    93.89 1355081265     0.00     0.00  UsedInBox(int (*) [25], int, int, int)
0.01   2167.56     0.17                             frame_dummy
0.00   2167.56     0.00        1     0.00     0.00  printGrid(int (*) [25])
```
```Call graph

granularity: each sample hit covers 2 byte(s) for 0.00% of 2085.44 seconds

index % time    self  children    called     name
<spontaneous>
[1]    100.0    0.00 2085.30                 main [1]
97.03 1988.27       1/1           SolveSudoku(int (*) [25]) [2]
0.00    0.00       1/1           printGrid(int (*) [25]) [14]
-----------------------------------------------
876012757             SolveSudoku(int (*) [25]) [2]
97.03 1988.27       1/1           main [1]
[2]    100.0   97.03 1988.27       1+876012757 SolveSudoku(int (*) [25]) [2]
101.19 1361.55 425478951/425478951     isSafe(int (*) [25], int, int, int) [3]
525.53    0.00 876012758/876012758     FindUnassignedLocation(int (*) [25], int&, int&) [5]
876012757             SolveSudoku(int (*) [25]) [2]
-----------------------------------------------
101.19 1361.55 425478951/425478951     SolveSudoku(int (*) [25]) [2]
[3]     70.1  101.19 1361.55 425478951         isSafe(int (*) [25], int, int, int) [3]
1011.03    0.00 425478951/425478951     UsedInRow(int (*) [25], int, int) [4]
259.56    0.00 590817023/590817023     UsedInCol(int (*) [25], int, int) [6]
90.96    0.00 1355081265/1355081265     UsedInBox(int (*) [25], int, int, int) [7]
-----------------------------------------------
1011.03    0.00 425478951/425478951     isSafe(int (*) [25], int, int, int) [3]
[4]     48.5 1011.03    0.00 425478951         UsedInRow(int (*) [25], int, int) [4]
-----------------------------------------------
525.53    0.00 876012758/876012758     SolveSudoku(int (*) [25]) [2]
[5]     25.2  525.53    0.00 876012758         FindUnassignedLocation(int (*) [25], int&, int&) [5]
-----------------------------------------------
259.56    0.00 590817023/590817023     isSafe(int (*) [25], int, int, int) [3]
[6]     12.4  259.56    0.00 590817023         UsedInCol(int (*) [25], int, int) [6]
-----------------------------------------------
90.96    0.00 1355081265/1355081265     isSafe(int (*) [25], int, int, int) [3]
[7]      4.4   90.96    0.00 1355081265         UsedInBox(int (*) [25], int, int, int) [7]
-----------------------------------------------
<spontaneous>
[8]      0.0    0.14    0.00                 frame_dummy [8]
-----------------------------------------------
0.00    0.00       1/1           main [1]
[14]     0.0    0.00    0.00       1         printGrid(int (*) [25]) [14]
-----------------------------------------------
Index by function name
[2] SolveSudoku(int (*) [25]) [7] UsedInBox(int (*) [25], int, int, int) [14] printGrid(int (*) [25])
[5] FindUnassignedLocation(int (*) [25], int&, int&) [6] UsedInCol(int (*) [25], int, int) [8] frame_dummy
[3] isSafe(int (*) [25], int, int, int) [4] UsedInRow(int (*) [25], int, int)
```

Assignment 1: EasyBMP

EasyBMP Bitmap image library (Sample Program: Image to black and white renderer)

Library: http://easybmp.sourceforge.net/

Sample code:
```/**/
#include "EasyBMP.h"

using namespace std;
int main(int argc, char* argv[]) {
// Create a new Bitmap image with EasyBMP
BMP Background;
Background.ReadFromFile(argv[1]);
BMP Output;
int picWidth = Background.TellWidth();
int picHeight = Background.TellHeight();
Output.SetSize(Background.TellWidth(), Background.TellHeight());
Output.SetBitDepth(1);
for (int i = 1; i < picWidth - 1; ++i) {
for (int j = 1; j < picHeight - 1; ++j) {
int col = (Background(i, j)->Blue + Background(i, j)->Green + 10 * Background(i, j)->Red) / 12;
if (col > 127) {
Output(i, j)->Red = 255;
Output(i, j)->Blue = 255;
Output(i, j)->Green = 255;
}
else {
Output(i, j)->Red = 0;
Output(i, j)->Blue = 0;
Output(i, j)->Green = 0;
}
}
}
Output.WriteToFile(argv[2]);
return 0;
}
/**/
```

The program was compiled using the following commands:

```g++ -c -pg -g BW.cpp EasyBMP.cpp
g++ -pg BW.o EasyBMP.o -o BW
rm *.o
```

Attempted to run the program with a number of files (8K resolution):

Sample Images

Flat profile (Cabin):
```Each sample counts as 0.01 seconds.
%   cumulative   self              self     total
time   seconds   seconds    calls   s/call   s/call  name
31.38      1.74     1.74 33177600     0.00     0.00  BMP::FindClosestColor(RGBApixel&)
23.52      3.04     1.30 198921624    0.00     0.00  BMP::operator()(int, int)
9.95      3.59     0.55        2     0.28     0.28  BMP::SetSize(int, int)
7.60      4.01     0.42 41663472     0.00     0.00  BMP::GetColor(int)
6.87      4.39     0.38                             main
5.43      4.69     0.30 74841076     0.00     0.00  IntPow(int, int)
3.62      4.89     0.20 74841072     0.00     0.00  BMP::TellNumberOfColors()
3.35      5.07     0.19     4320     0.00     0.00  BMP::Write1bitRow(unsigned char*, int, int)
2.53      5.21     0.14 124990416    0.00     0.00  IntSquare(int)
2.17      5.33     0.12     4320     0.00     0.00  BMP::Read24bitRow(unsigned char*, int, int)
1.63      5.42     0.09        2     0.05     0.05  BMP::~BMP()
0.90      5.47     0.05                             GetEasyBMPwarningState()
0.72      5.51     0.04        1     0.04     0.04  _GLOBAL__sub_I_EasyBMPwarnings
0.18      5.52     0.01        2     0.01     0.01  BMP::TellWidth()
0.18      5.53     0.01        1     0.01     2.99  BMP::WriteToFile(char const*)
0.00      5.53     0.00       16     0.00     0.00  SafeFread(char*, int, int, _IO_FILE*)
0.00      5.53     0.00        6     0.00     0.00  IsBigEndian()
0.00      5.53     0.00        2     0.00     0.00  EasyBMPcheckDataSize()
0.00      5.53     0.00        2     0.00     0.00  BMP::TellHeight()
0.00      5.53     0.00        2     0.00     0.00  BMP::SetBitDepth(int)
0.00      5.53     0.00        2     0.00     0.00  BMP::BMP()
0.00      5.53     0.00        2     0.00     0.00  BMFH::BMFH()
0.00      5.53     0.00        2     0.00     0.00  BMIH::BMIH()
0.00      5.53     0.00        1     0.00     0.00  _GLOBAL__sub_I_main
0.00      5.53     0.00        1     0.00     0.00  __static_initialization_and_destruction_0(int, int)
0.00      5.53     0.00        1     0.00     0.00  __static_initialization_and_destruction_0(int, int)
0.00      5.53     0.00        1     0.00     0.40  BMP::ReadFromFile(char const*)
0.00      5.53     0.00        1     0.00     0.00  BMP::CreateStandardColorTable()
```
Flat profile (Lake):
```Each sample counts as 0.01 seconds.
%   cumulative   self              self     total
time   seconds   seconds    calls   s/call   s/call  name
30.60      1.71     1.71 33177600     0.00     0.00  BMP::FindClosestColor(RGBApixel&)
21.12      2.89     1.18 198921624    0.00     0.00  BMP::operator()(int, int)
10.20      3.46     0.57        2     0.29     0.29  BMP::SetSize(int, int)
8.59      3.94     0.48                             main
6.26      4.29     0.35 76982189     0.00     0.00  IntPow(int, int)
6.08      4.63     0.34 43804585     0.00     0.00  BMP::GetColor(int)
5.55      4.94     0.31 76982185     0.00     0.00  BMP::TellNumberOfColors()
3.76      5.15     0.21     4320     0.00     0.00  BMP::Write1bitRow(unsigned char*, int, int)
3.76      5.36     0.21 131413755    0.00     0.00  IntSquare(int)
2.15      5.48     0.12     4320     0.00     0.00  BMP::Read24bitRow(unsigned char*, int, int)
1.07      5.54     0.06        2     0.03     0.03  BMP::~BMP()
0.54      5.57     0.03                             GetEasyBMPwarningState()
0.36      5.59     0.02        1     0.02     0.02  _GLOBAL__sub_I_EasyBMPwarnings
0.00      5.59     0.00       16     0.00     0.00  SafeFread(char*, int, int, _IO_FILE*)
0.00      5.59     0.00        6     0.00     0.00  IsBigEndian()
0.00      5.59     0.00        2     0.00     0.00  EasyBMPcheckDataSize()
0.00      5.59     0.00        2     0.00     0.00  BMP::TellHeight()
0.00      5.59     0.00        2     0.00     0.00  BMP::SetBitDepth(int)
0.00      5.59     0.00        2     0.00     0.00  BMP::TellWidth()
0.00      5.59     0.00        2     0.00     0.00  BMP::BMP()
0.00      5.59     0.00        2     0.00     0.00  BMFH::BMFH()
0.00      5.59     0.00        2     0.00     0.00  BMIH::BMIH()
0.00      5.59     0.00        1     0.00     0.00  _GLOBAL__sub_I_main
0.00      5.59     0.00        1     0.00     0.00  __static_initialization_and_destruction_0(int, int)
0.00      5.59     0.00        1     0.00     0.00  __static_initialization_and_destruction_0(int, int)
0.00      5.59     0.00        1     0.00     3.13  BMP::WriteToFile(char const*)
0.00      5.59     0.00        1     0.00     0.41  BMP::ReadFromFile(char const*)
0.00      5.59     0.00        1     0.00     0.00  BMP::CreateStandardColorTable()
```

Assignment 1: Julia Sets

This portion of the assignment focuses on Julia sets with the quadratic formula:

``` fc(z) = z^2 + c;
Where c and z are complex numbers
```

Psuedo code

``` for(Pixel pix in image){
pix.color = colorFunction(escapeValue(pix.loc, julia));
}
```
``` escapeValue(Complex loc, Complex julia){
int cycles = 0;
while(|loc| <=2 && ++cycles < MAXCYCLES){
loc = loc * loc + julia;
}
return cycles;
}
```
Julia.cpp
``` /**
* Julia.cpp
* Vincent Terpstra
* Feb 18 / 2019
* Functions to display a Julia set
*/
```
``` #include <iostream>
#include <math.h>
#include "Bitmap.h"
```
``` #define MAXCYCLES 1000
#define ITER 1
#define PI 3.14159
```
``` void calcJulia(int* array, int width, int height, float jR, float jI);
void createBMP(int* array, int width, int height);
```
``` int main(int size, const char ** args) {
int height = (size > 1) ? atoi(args[1]) : 1000;
int width = height / 2 * 3;
```
```     int * array = new int[height * width];
//Initial Julia Point value
float jI = -0.4f;
float jR =  0.6f;
//for rotating the Julia point around the origin
float angle = 2.0f / ITER * PI;
float cosA = cos(angle);
float sinA = sin(angle);
for(int i = 0; i < ITER; i++){
calcJulia(array, width, height, jI, jR);
createBMP(array, width, height);

//rotate the Julia point
float tmp = jI;
jI = cosA * jI - sinA * jR;
jR = cosA * jR + sinA * tmp;
}

delete[] array;
return 0;
}
/**
* calcDepth
*      finds the escape value of the Julia set
*      inputs sR, sI: starting position on the map
*          jR, jI: values of the julia set
*      exits when |s| > 2 or reached MAXCYCLES
*      return the escape value or MAXCYCLES
*/
int calcDepth(float sR, float sI, const float jR, const float jI){
int cycle = 0;
while(sR * sR + sI * sI <= 4 && ++cycle < MAXCYCLES){
// s(next) = s * s + j
float tmp = sR;
sR = sR * sR - sI * sI + jR;
sI = 2 * tmp * sI + jI;
}
return cycle;
}
```
``` /**
* calcJulia
*  Function to calculate the values from (-1.5, 1.5),(-1, 1) in the Julia set
*      saves the values onto the array
*/
void calcJulia(int * array, int width, int height, const float jR, const float jI){
float delta = 2 / (float)height;
float x, y = -1;
for(int i = 0; i < height; i++){
x = -1.5f;
for(int j = 0; j < width; j++){
*array++ = calcDepth(x, y, jR, jI);
x+=delta;
}
y += delta;
}
}
```
``` struct Pix {
float val;
int r, g, b;
Pix(float val, int r, int g, int b) :
val(val), r(r), g(g), b(b) {}
};
```
``` bool lerp(float clr, Pix& first, Pix& second, Bitmap & map);
```
``` /**
* createBMP
*  maps the values in the array to a .bmp file
*  linear interpolates between color values
*/
void createBMP(int * array, int width, int height){
static int idx = 0;
char name[] = "julia000.bmp";
name[5] = (idx / 100) + '0';
name[6] = ((idx / 10) % 10) + '0';
name[7] = (idx % 10) + '0';
Bitmap map (name, width, height);
//idx++;
const int pix = width * height;
struct Pix
clr0(0, 20, 20, 30),
clr1(10, 20, 10, 100),
clr2(180, 20, 170, 205),
clr3(MAXCYCLES, 0, 0, 0);
const int * pMax = array + width * height;
while(array < pMax){
float clr = *array++;
bool draw =
lerp(clr, clr0, clr1, map) ||
lerp(clr, clr1, clr2, map) ||
lerp(clr, clr2, clr3, map);
}
}
```
``` /**
* Function to linearly interpolate color values and draw to the Bitmap
*/
bool lerp(float clr, Pix& first, Pix& second, Bitmap & map){
if(clr <= second.val){
float diff = (clr - first.val) / (second.val - first.val);
map.addColor(
first.b + diff * (second.b - first.b),
first.g + diff * (second.g - first.g),
first.r + diff * (second.r - first.r)
);
return true;
}
return false;
}
```

To view the full c++ code github link

``` This code is tested using the parameters
Range R(-1.5, 1.5) I(-1, 1)
MAXCYCLES 1000
Julia values = .72 * e^(i*θ): θ[0, 2π] : 100 intervals
```
Flat Profiles
```Width: 750 Height: 500
Flat profile:
Each sample counts as 0.01 seconds.
%   cumulative   self              self     total
time   seconds   seconds    calls  ns/call  ns/call  name
96.00     20.87    20.87                             calcJulia(int*, int, int, float, float)
2.08     21.32     0.45 37500000    12.04    12.04  Bitmap::addColor(int, int, int)
1.75     21.70     0.38 49447183     7.71    16.84  lerp(float, Pix&, Pix&, Bitmap&)
0.46     21.80     0.10                             createBMP(int*, int, int)
0.00     21.80     0.00      100     0.00     0.00  generateBitmapImage(unsigned char*, int, int, char const*)
0.00     21.80     0.00      100     0.00     0.00  createBitmapFileHeader(int, int, int)
0.00     21.80     0.00      100     0.00     0.00  createBitmapInfoHeader(int, int)
0.00     21.80     0.00      100     0.00     0.00  Bitmap::Bitmap(char const*, int, int)
0.00     21.80     0.00      100     0.00     0.00  Bitmap::~Bitmap()
0.00     21.80     0.00        1     0.00     0.00  _GLOBAL__sub_I_main
```
```Width: 1500 Height: 1000
Flat profile:
Each sample counts as 0.01 seconds.
%   cumulative   self              self     total
time   seconds   seconds    calls  us/call  us/call  name
97.05     84.67    84.67                             calcJulia(int*, int, int, float, float)
1.66     86.12     1.45 150000000     0.01     0.01  Bitmap::addColor(int, int, int)
1.12     87.09     0.97 197791886     0.00     0.01  lerp(float, Pix&, Pix&, Bitmap&)
0.37     87.41     0.32                             createBMP(int*, int, int)
0.01     87.42     0.01      100    50.17    50.17  Bitmap::Bitmap(char const*, int, int)
0.00     87.42     0.00      100     0.00     0.00  generateBitmapImage(unsigned char*, int, int, char const*)
0.00     87.42     0.00      100     0.00     0.00  createBitmapFileHeader(int, int, int)
0.00     87.42     0.00      100     0.00     0.00  createBitmapInfoHeader(int, int)
0.00     87.42     0.00      100     0.00     0.00  Bitmap::~Bitmap()
0.00     87.42     0.00        1     0.00     0.00  _GLOBAL__sub_I_main
```
``` Width: 2250 Height: 1500
Flat profile:
Each sample counts as 0.01 seconds.
%   cumulative   self              self     total
time   seconds   seconds    calls  us/call  us/call  name
96.39    189.55   189.55                             calcJulia(int*, int, int, float, float)
1.89    193.27     3.72 337500000     0.01     0.01  Bitmap::addColor(int, int, int)
1.49    196.20     2.93 445028641     0.01     0.01  lerp(float, Pix&, Pix&, Bitmap&)
0.45    197.09     0.89                             createBMP(int*, int, int)
0.01    197.11     0.02      100   150.51   150.51  Bitmap::Bitmap(char const*, int, int)
0.00    197.11     0.00      100     0.00     0.00  generateBitmapImage(unsigned char*, int, int, char const*)
0.00    197.11     0.00      100     0.00     0.00  createBitmapFileHeader(int, int, int)
0.00    197.11     0.00      100     0.00     0.00  createBitmapInfoHeader(int, int)
0.00    197.11     0.00      100     0.00     0.00  Bitmap::~Bitmap()
0.00    197.11     0.00        1     0.00     0.00  _GLOBAL__sub_I_main
```
``` Width: 3000 Height: 2000
Flat profile:
Each sample counts as 0.01 seconds.
%   cumulative   self              self     total
time   seconds   seconds    calls  us/call  us/call  name
95.74    332.96   332.96                             calcJulia(int*, int, int, float, float)
2.79    342.68     9.72 600000000     0.02     0.02  Bitmap::addColor(int, int, int)
1.28    347.14     4.46 791167185     0.01     0.02  lerp(float, Pix&, Pix&, Bitmap&)
0.44    348.66     1.52                             createBMP(int*, int, int)
0.01    348.69     0.03      100   250.84   250.84  Bitmap::Bitmap(char const*, int, int)
0.00    348.69     0.00      100     0.00     0.00  generateBitmapImage(unsigned char*, int, int, char const*)
0.00    348.69     0.00      100     0.00     0.00  createBitmapFileHeader(int, int, int)
0.00    348.69     0.00      100     0.00     0.00  createBitmapInfoHeader(int, int)
0.00    348.69     0.00      100     0.00     0.00  Bitmap::~Bitmap()
0.00    348.69     0.00        1     0.00     0.00  _GLOBAL__sub_I_main
```
Call Graphs
```Width: 750 Height: 500
Call graph
```
```granularity: each sample hit covers 2 byte(s) for 0.05% of 21.80 seconds
index % time    self  children    called     name
<spontaneous>
[1]     96.0   20.93    0.00                 calcJulia(int*, int, int, float, float) [1]
-----------------------------------------------
<spontaneous>
[2]      4.0    0.14    0.73                 createBMP(int*, int, int) [2]
0.33    0.40 49447183/49447183     lerp(float, Pix&, Pix&, Bitmap&) [3]
0.00    0.00     100/100         Bitmap::Bitmap(char const*, int, int) [15]
0.00    0.00     100/100         Bitmap::~Bitmap() [16]
-----------------------------------------------
0.33    0.40 49447183/49447183     createBMP(int*, int, int) [2]
[3]      3.4    0.33    0.40 49447183         lerp(float, Pix&, Pix&, Bitmap&) [3]
0.40    0.00 37500000/37500000     Bitmap::addColor(int, int, int) [4]
-----------------------------------------------
0.40    0.00 37500000/37500000     lerp(float, Pix&, Pix&, Bitmap&) [3]
[4]      1.8    0.40    0.00 37500000         Bitmap::addColor(int, int, int) [4]
-----------------------------------------------
0.00    0.00     100/100         Bitmap::~Bitmap() [16]
[12]     0.0    0.00    0.00     100         generateBitmapImage(unsigned char*, int, int, char const*)[12]
0.00    0.00     100/100         createBitmapFileHeader(int, int, int) [13]
0.00    0.00     100/100         createBitmapInfoHeader(int, int) [14]
-----------------------------------------------
0.00    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*) [12]
[13]     0.0    0.00    0.00     100         createBitmapFileHeader(int, int, int) [13]
-----------------------------------------------
0.00    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*) [12]
[14]     0.0    0.00    0.00     100         createBitmapInfoHeader(int, int) [14]
-----------------------------------------------
0.00    0.00     100/100         createBMP(int*, int, int) [2]
[15]     0.0    0.00    0.00     100         Bitmap::Bitmap(char const*, int, int) [15]
-----------------------------------------------
0.00    0.00     100/100         createBMP(int*, int, int) [2]
[16]     0.0    0.00    0.00     100         Bitmap::~Bitmap() [16]
0.00    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*) [12]
-----------------------------------------------
0.00    0.00       1/1           __libc_csu_init [25]
[17]     0.0    0.00    0.00       1         _GLOBAL__sub_I_main [17]
-----------------------------------------------
Index by function name
[17] _GLOBAL__sub_I_main (julia.cpp) [3] lerp(float, Pix&, Pix&, Bitmap&) [15] Bitmap::Bitmap(char const*, int, int)
[12] generateBitmapImage(unsigned char*, int, int, char const*) [1] calcJulia(int*, int, int, float,  float) [16] Bitmap::~Bitmap()
[13] createBitmapFileHeader(int, int, int) [2] createBMP(int*, int, int)
[14] createBitmapInfoHeader(int, int) [4] Bitmap::addColor(int, int, int)
```

```Width: 1500 Height: 1000
Call graph
```
```granularity: each sample hit covers 2 byte(s) for 0.01% of 87.23 seconds
index % time    self  children    called     name
<spontaneous>
[1]     96.0   83.70    0.00                 calcJulia(int*, int, int, float, float) [1]
-----------------------------------------------
<spontaneous>
[2]      4.0    0.37    3.16                 createBMP(int*, int, int) [2]
1.39    1.76 197791886/197791886     lerp(float, Pix&, Pix&, Bitmap&) [3]
0.01    0.00     100/100         Bitmap::Bitmap(char const*, int, int) [5]
0.00    0.00     100/100         Bitmap::~Bitmap() [16]
-----------------------------------------------
1.39    1.76 197791886/197791886     createBMP(int*, int, int) [2]
[3]      3.6    1.39    1.76 197791886         lerp(float, Pix&, Pix&, Bitmap&) [3]
1.76    0.00 150000000/150000000     Bitmap::addColor(int, int, int) [4]
-----------------------------------------------
1.76    0.00 150000000/150000000     lerp(float, Pix&, Pix&, Bitmap&) [3]
[4]      2.0    1.76    0.00 150000000         Bitmap::addColor(int, int, int) [4]
-----------------------------------------------
0.01    0.00     100/100         createBMP(int*, int, int) [2]
[5]      0.0    0.01    0.00     100         Bitmap::Bitmap(char const*, int, int) [5]
-----------------------------------------------
0.00    0.00     100/100         Bitmap::~Bitmap() [16]
[13]     0.0    0.00    0.00     100         generateBitmapImage(unsigned char*, int, int, char const*) [13]
0.00    0.00     100/100         createBitmapFileHeader(int, int, int) [14]
0.00    0.00     100/100         createBitmapInfoHeader(int, int) [15]
-----------------------------------------------
0.00    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*) [13]
[14]     0.0    0.00    0.00     100         createBitmapFileHeader(int, int, int) [14]
-----------------------------------------------
0.00    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*) [13]
[15]     0.0    0.00    0.00     100         createBitmapInfoHeader(int, int) [15]
-----------------------------------------------
0.00    0.00     100/100         createBMP(int*, int, int) [2]
[16]     0.0    0.00    0.00     100         Bitmap::~Bitmap() [16]
0.00    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*)  [13]
-----------------------------------------------
0.00    0.00       1/1           __libc_csu_init [25]
[17]     0.0    0.00    0.00       1         _GLOBAL__sub_I_main [17]
-----------------------------------------------
Index by function name
[17] _GLOBAL__sub_I_main (julia.cpp) [3] lerp(float, Pix&, Pix&, Bitmap&) [5] Bitmap::Bitmap(char const*, int, int)
[13] generateBitmapImage(unsigned char*, int, int, char const*) [1] calcJulia(int*, int, int, float, float) [16] Bitmap::~Bitmap()
[14] createBitmapFileHeader(int, int, int) [2] createBMP(int*, int, int)
[15] createBitmapInfoHeader(int, int) [4] Bitmap::addColor(int, int, int)
```

```Width: 2250 Height: 1500
Call graph
```
```granularity: each sample hit covers 2 byte(s) for 0.01% of 196.08 seconds
index % time    self  children    called     name
<spontaneous>
[1]     96.0  188.22    0.00                 calcJulia(int*, int, int, float, float) [1]
-----------------------------------------------
<spontaneous>
[2]      4.0    0.95    6.91                 createBMP(int*, int, int) [2]
2.78    4.12 445028641/445028641     lerp(float, Pix&, Pix&, Bitmap&) [3]
0.00    0.01     100/100         Bitmap::~Bitmap() [6]
0.01    0.00     100/100         Bitmap::Bitmap(char const*, int, int) [7]
-----------------------------------------------
2.78    4.12 445028641/445028641     createBMP(int*, int, int) [2]
[3]      3.5    2.78    4.12 445028641         lerp(float, Pix&, Pix&, Bitmap&) [3]
4.12    0.00 337500000/337500000     Bitmap::addColor(int, int, int) [4]
-----------------------------------------------
4.12    0.00 337500000/337500000     lerp(float, Pix&, Pix&, Bitmap&) [3]
[4]      2.1    4.12    0.00 337500000         Bitmap::addColor(int, int, int) [4]
-----------------------------------------------
0.01    0.00     100/100         Bitmap::~Bitmap() [6]
[5]      0.0    0.01    0.00     100         generateBitmapImage(unsigned char*, int, int, char const*) [5]
0.00    0.00     100/100         createBitmapFileHeader(int, int, int) [15]
0.00    0.00     100/100         createBitmapInfoHeader(int, int) [16]
-----------------------------------------------
0.00    0.01     100/100         createBMP(int*, int, int) [2]
[6]      0.0    0.00    0.01     100         Bitmap::~Bitmap() [6]
0.01    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*)  [5]
-----------------------------------------------
0.01    0.00     100/100         createBMP(int*, int, int) [2]
[7]      0.0    0.01    0.00     100         Bitmap::Bitmap(char const*, int, int) [7]
-----------------------------------------------
0.00    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*)  [5]
[15]     0.0    0.00    0.00     100         createBitmapFileHeader(int, int, int) [15]
-----------------------------------------------
0.00    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*)  [5]
[16]     0.0    0.00    0.00     100         createBitmapInfoHeader(int, int) [16]
-----------------------------------------------
0.00    0.00       1/1           __libc_csu_init [25]
[17]     0.0    0.00    0.00       1         _GLOBAL__sub_I_main [17]
-----------------------------------------------
Index by function name
[17] _GLOBAL__sub_I_main (julia.cpp) [3] lerp(float, Pix&, Pix&, Bitmap&) [7] Bitmap::Bitmap(char const*, int, int)
[5] generateBitmapImage(unsigned char*, int, int, char const*) [1] calcJulia(int*, int, int, float, float) [6] Bitmap::~Bitmap()
[15] createBitmapFileHeader(int, int, int) [2] createBMP(int*, int, int)
[16] createBitmapInfoHeader(int, int) [4] Bitmap::addColor(int, int, int)
```
```Width: 3000 Height: 2000
Call graph
```
```granularity: each sample hit covers 2 byte(s) for 0.00% of 347.05 seconds
index % time    self  children    called     name
<spontaneous>
[1]     96.1  333.37    0.00                 calcJulia(int*, int, int, float, float) [1]
-----------------------------------------------
<spontaneous>
[2]      3.9    1.21   12.47                 createBMP(int*, int, int) [2]
5.74    6.73 791167185/791167185     lerp(float, Pix&, Pix&, Bitmap&) [3]
0.01    0.00     100/100         Bitmap::Bitmap(char const*, int, int) [5]
0.00    0.00     100/100         Bitmap::~Bitmap() [16]
-----------------------------------------------
5.74    6.73 791167185/791167185     createBMP(int*, int, int) [2]
[3]      3.6    5.74    6.73 791167185         lerp(float, Pix&, Pix&, Bitmap&) [3]
6.73    0.00 600000000/600000000     Bitmap::addColor(int, int, int) [4]
-----------------------------------------------
6.73    0.00 600000000/600000000     lerp(float, Pix&, Pix&, Bitmap&) [3]
[4]      1.9    6.73    0.00 600000000         Bitmap::addColor(int, int, int) [4]
-----------------------------------------------
0.01    0.00     100/100         createBMP(int*, int, int) [2]
[5]      0.0    0.01    0.00     100         Bitmap::Bitmap(char const*, int, int) [5]
-----------------------------------------------
0.00    0.00     100/100         Bitmap::~Bitmap() [16]
[13]     0.0    0.00    0.00     100         generateBitmapImage(unsigned char*, int, int, char const*)  [13]
0.00    0.00     100/100         createBitmapFileHeader(int, int, int) [14]
0.00    0.00     100/100         createBitmapInfoHeader(int, int) [15]
-----------------------------------------------
0.00    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*)  [13]
[14]     0.0    0.00    0.00     100         createBitmapFileHeader(int, int, int) [14]
-----------------------------------------------
0.00    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*)  [13]
[15]     0.0    0.00    0.00     100         createBitmapInfoHeader(int, int) [15]
-----------------------------------------------
0.00    0.00     100/100         createBMP(int*, int, int) [2]
[16]     0.0    0.00    0.00     100         Bitmap::~Bitmap() [16]
0.00    0.00     100/100         generateBitmapImage(unsigned char*, int, int, char const*)  [13]
-----------------------------------------------
0.00    0.00       1/1           __libc_csu_init [25]
[17]     0.0    0.00    0.00       1         _GLOBAL__sub_I_main [17]
-----------------------------------------------
Index by function name
[17] _GLOBAL__sub_I_main (julia.cpp) [3] lerp(float, Pix&, Pix&, Bitmap&) [5] Bitmap::Bitmap(char const*, int, int)
[13] generateBitmapImage(unsigned char*, int, int, char const*) [1] calcJulia(int*, int, int, float, float) [16] Bitmap::~Bitmap()
[14] createBitmapFileHeader(int, int, int) [2] createBMP(int*, int, int)
[15] createBitmapInfoHeader(int, int) [4] Bitmap::addColor(int, int, int)
```
Generated Image of Julia set at (-0.4, 0.6)

Assignment 1: Selection for parallelizing

After reviewing the three programs above, we decided to attempt to parallelize the Sudoku Solver Program for a few reasons.

```1.  By increasing the dimensions of the smaller matrices that make up a sudoku by one, we see a major increase in the time it takes to solve the sudoku, from almost instantly to around 38 seconds, and then to 36 minutes. With a 25x25 sudoku (of 5x5 matrices), several functions were called over 100 million times.
```
```2. Based on the massive time increases and similarity to the Hamiltonian Path Problem [2] which also uses backtracking to find a solution, we believe the run time of the sudoku solver to have a Big O notation that approaches O(n!) where 'n' is the number of blank spaces in the sudoku as the sudoku solver uses recursion to check every single possible solution, returning to previous steps if the tried solution does not work. O(n!) is an even worse runtime than O(n^2).
```
```3.  The Julia sets still took less than 6 minutes after increasing the image size, and the EasyBMP only took a few seconds to convert a large, high resolution image.  Therefore, the Sudoku Solver had the greatest amount of time to be shaven off through optimization and thus offered the most challenge.
```

Assignment 2

Code for Solving a Sudoku using backtracking
```#include <stdio.h>
#include <iostream>
#include <cstdlib>
#include <ctime>
#include <iomanip>
// CUDA header file
#include "cuda_runtime.h"
#include "device_launch_parameters.h"
// A Backtracking program  in C++ to solve Sudoku problem
#include <stdio.h>
// UNASSIGNED is used for empty cells in sudoku grid
#define UNASSIGNED 0
// N is used for the size of Sudoku grid. Size will be NxN
#define BOXWIDTH 3
#define N 9
// This function finds an entry in grid that is still unassigned
bool FindUnassignedLocation(int grid[N][N], int &row, int &col);
// Checks whether it will be legal to assign num to the given row, col
__global__ void makeGuess(int* d_a, int guess, int row, int col) {
__shared__ bool found;
bool control = !(threadIdx.x || threadIdx.y);
int tidx = 0;
if (threadIdx.y == 2) { //BOXES
tidx = (((row / BOXWIDTH) * BOXWIDTH) + threadIdx.x / BOXWIDTH) * N
+ threadIdx.x % BOXWIDTH + (col / BOXWIDTH) * BOXWIDTH;
} else if (threadIdx.y == 1) { //ROWS
tidx = row * N + threadIdx.x;
} else { //COLUMNS
tidx = threadIdx.x * N + col;
}
if (control)
found = true;
__syncthreads();
while (found) {
__syncthreads();
if (control)
found = false;
__syncthreads();
if (d_a[tidx] == ++guess)
found = true;
__syncthreads();
}
if (control)
d_a[row * N + col] = guess;
}
```
```/* Takes a partially filled-in grid and attempts to assign values to
all unassigned locations in such a way to meet the requirements
for Sudoku solution (non-duplication across rows, columns, and boxes) */
bool SolveSudoku(int grid[N][N], int* d_a, int row, int col)
{
// If there is no unassigned location, we are done
if (!FindUnassignedLocation(grid, row, col))
return true; // success!
dim3 block(N, 3);
int guess = 0;
while (guess <= N) {
makeGuess << <1, block >> > (d_a, guess, row, col);
cudaMemcpy(&guess, d_a + row * N + col, sizeof(int), cudaMemcpyDeviceToHost);
if (guess <= N && SolveSudoku(grid, d_a, row, col + 1))
return true;
}
//Erase the guess on the host
int zero = UNASSIGNED;
cudaMemcpy(d_a + row * N + col, &zero, sizeof(int), cudaMemcpyHostToDevice);
return false; // this triggers backtracking
}
```
```/* Searches the grid to find an entry that is still unassigned. If
found, the reference parameters row, col will be set the location
that is unassigned, and true is returned. If no unassigned entries
remain, false is returned. */
bool FindUnassignedLocation(int grid[N][N], int &row, int &col)
{
for (; row < N; ++row) {
for (; col < N; ++col)
if (grid[row][col] == UNASSIGNED)
return true;
col = 0;
}
return false;
}
```
```/* A utility function to print grid  */
void printGrid(int grid[N][N])
{
for (int row = 0; row < N; row++){
for (int col = 0; col < N; col++)
printf("%2d", grid[row][col]);
printf("\n");
}
}
```
```/* Driver Program to test above functions */
int main()
{
```
```   /* 0 means unassigned cells */
int grid[N][N] =
{{3, 0, 6, 5, 0, 8, 4, 0, 0},
{5, 2, 0, 0, 0, 0, 0, 0, 0},
{0, 8, 7, 0, 0, 0, 0, 3, 1},
{0, 0, 3, 0, 1, 0, 0, 8, 0},
{9, 0, 0, 8, 6, 3, 0, 0, 5},
{0, 5, 0, 0, 9, 0, 6, 0, 0},
{1, 3, 0, 0, 0, 0, 2, 5, 0},
{0, 0, 0, 0, 0, 0, 0, 7, 4},
{0, 0, 5, 2, 0, 6, 3, 0, 0} };
```
```   /**
int grid[N][N] =
{{0,  8,   0,  0,  0,  0,  0,  3,  0,  0,  0, 10,  9,  7, 11, 0},
{0,  9,  15, 13,  0, 10,  0,  0,  2,  6,  8, 16,  0,  0,  0, 0},
{0,  0,  16,  0, 15,  0,  8,  0,  9,  0,  0,  0,  6,  0,  2, 0},
{1,  0,   2,  0,  9, 11,  4,  6, 15,  3,  5,  7,  0,  0, 12, 0},
{16, 6,   4,  0,  5,  2,  0,  0,  1,  0,  0,  0, 11,  0,  0, 12},
{5,  11,  0,  0,  0,  3,  0, 15,  0, 16,  0, 13,  0,  1,  0, 8},
{0,  0,   3,  0,  0,  6, 11, 14,  0,  5,  7,  0,  0,  9,  0, 0},
{0,  0,   0, 14,  8,  0, 10,  0,  0, 11, 12,  0,  0,  0,  0, 0},
{0,  7,  13,  0,  0,  0,  0, 12,  0,  8,  9,  0,  0,  0,  3, 0},
{0,  0,  11,  9,  0,  7,  0,  0,  0,  0,  0, 12,  0,  8, 16, 5},
{0,  0,  10,  0, 11, 13,  0,  0,  0,  0,  0,  3, 12,  0,  6, 0},
{0,  5,   0,  0, 10, 15,  0,  1,  7,  2,  0,  0, 14, 11,  0, 0},
{0,  0,   5,  0,  0, 12, 14,  0,  0, 10,  0,  0, 15,  0,  0, 4},
{9,  0,  14,  6,  0,  0,  1,  0, 16,  0,  2,  0,  3,  0, 13, 0},
{8,  13,  0,  4,  0,  0,  0,  0, 12,  7,  3,  0,  0,  6,  0, 0},
{0,  16, 12,  0,  0,  5,  0,  9,  0, 13, 14,  4,  1,  0,  0, 0} };
/**/
```
```   int* d_a;
cudaMalloc((void**)&d_a, N*N * sizeof(int));
cudaMemcpy(d_a, grid, N*N * sizeof(int), cudaMemcpyHostToDevice);
SolveSudoku(grid, d_a, 0, 0);
cudaMemcpy(grid, d_a, N*N * sizeof(int), cudaMemcpyDeviceToHost);
printGrid(grid);
cudaFree(d_a);
return 0;
}
```

This code is capable of solving the 9x9 matrix supplied HOWEVER with the backtracking algorithm substituting values and the communications delay between the GPU and CPU, This code is unable to solve the 16x16 in any reasonable amount of time (I stopped it at 10+ minutes). If you consider the 130+ empty spaces in the grid I estimate over 130^2 calls to cudaMemcpy either way...

So we need an algorithm which will check each open spot, calculate all possible values which can fit there, and assign single values. We can also check each section (Box, row, col) for values which can only go in one place

Attempt One...
```/**
* Vincent Terpstra
* Sudoku.cu
* March 18 / 2019
* An Optimistic approach to solving a Sudoku on a CUDA enabled GPU
*    Assumes that the puzzle is deterministic(single solvable solution)
*        AND each next step can be found with the kernel
* KERNEL: educatedGuess
*   searches each square in a box for
*    squares that have only a single appropiate value
*    OR values that (in the box) can only fit in one square
*/
```
```#include <stdio.h>
#include <iostream>
#include <cstdlib>
#include <ctime>
#include <iomanip>
// CUDA header file
#include "cuda_runtime.h"
#include <device_launch_parameters.h>
#ifndef __CUDACC__
#define __CUDACC__
#endif
#include <device_functions.h>
#include <stdio.h>
// UNASSIGNED is used for empty cells in sudoku grid
#define UNASSIGNED 0
// N is used for the size of Sudoku grid. Size will be NxN
#define BOXWIDTH 5
#define N (BOXWIDTH * BOXWIDTH)
```
```/*
* kernel to solve a sudoku
* Input: sudoku puzzle partitioned into boxes
*	* d_a = the sudoku puzzle
*	figures out what values can fit in each square
*  figures out how many spots each value can go
*  assigns the appropiate values,
*	saves to addedIdx to show that there is a change
*/
```
```__global__ void educatedGuess(int * d_a, int * addedIdx) {
int idx   = threadIdx.x + BOXWIDTH * threadIdx.y;
int gridX = threadIdx.x + BOXWIDTH * blockIdx.x;
int gridY = threadIdx.y + BOXWIDTH * blockIdx.y;
int gridIdx = gridX + N * gridY;
__shared__ bool hasValue[N]; //If the value occurs in the box
__shared__ int  inBox[N];	 //Number of places each integer can go in the box
hasValue[idx] = false;
inBox[idx] = 0;
__syncthreads();
int at = d_a[gridIdx];
if (at != 0)
hasValue[at - 1] = true;
__syncthreads();
if (at != 0)
return;
//For remembering which values were seen in the rows and columns
bool foundVal[N];
for (int i = 0; i < N; ++i)
foundVal[i] = hasValue[i];

for (int check = 0; check < N; check++) {
foundVal[d_a[N * check + gridX] - 1] = true;
foundVal[d_a[N * gridY + check] - 1] = true;
}
int fndVals = 0;
for( int i = 0; i < N; ++i)
if (!foundVal[i]) {
fndVals++;
at = i + 1;
}
if (fndVals == 1) {
//Only one possible value for this index
d_a[gridIdx]  = at;        //assign value
addedIdx[0]   = gridIdx;   //to tell host that the table has changed
inBox[at - 1] = 4; //Prevent one index per value
}
__syncthreads();
//Calculate the number of places each integer can go in the box
for (int i = 0; i < N; ++i) {
int num = (idx + i) % N; //keep each thread on a seperate idx
if (!foundVal[num])
inBox[num]++;
__syncthreads();
}
for (int i = 0; i < N; ++i) {
//if there is only one possible index for that value assign the value
if (inBox[i] == 1 && !foundVal[i]) {
d_a[gridIdx] = i + 1;    //assign value
addedIdx[0] = gridIdx;   //to tell host that the table has changed
}
}
}
```
```/* Solves the Sudoku, with best values */
void SolveSudoku(int grid[N][N], int* d_a, int* d_results)
{
dim3 block(BOXWIDTH, BOXWIDTH);
int lastIdx(-1), nextIdx(-1);
do {
lastIdx = nextIdx;
educatedGuess << <block, block >> > (d_a, d_results);
cudaMemcpy(&nextIdx, d_results, sizeof(int), cudaMemcpyDeviceToHost);
} while (lastIdx != nextIdx);
}
```
```/* A utility function to print grid  */
void printGrid(int grid[N][N])
{ for (int row = 0; row < N; row++) {
for (int col = 0; col < N; col++)
printf("%3d", grid[row][col]);
printf("\n");
}
}
```
```/* Driver Program to test above functions */
int main()
{ /* 0 means unassigned cells *
int grid[N][N] =
{ {3, 0, 6, 5, 0, 8, 4, 0, 0},
{5, 2, 0, 0, 0, 0, 0, 0, 0},
{0, 8, 7, 0, 0, 0, 0, 3, 1},
{0, 0, 3, 0, 1, 0, 0, 8, 0},
{9, 0, 0, 8, 6, 3, 0, 0, 5},
{0, 5, 0, 0, 9, 0, 6, 0, 0},
{1, 3, 0, 0, 0, 0, 2, 5, 0},
{0, 0, 0, 0, 0, 0, 0, 7, 4},
{0, 0, 5, 2, 0, 6, 3, 0, 0} };
```
```   /**
int grid[N][N] =
{{0,  8,   0,  0,  0,  0,  0,  3,  0,  0,  0, 10,  9,  7, 11, 0},
{0,  9,  15, 13,  0, 10,  0,  0,  2,  6,  8, 16,  0,  0,  0, 0},
{0,  0,  16,  0, 15,  0,  8,  0,  9,  0,  0,  0,  6,  0,  2, 0},
{1,  0,   2,  0,  9, 11,  4,  6, 15,  3,  5,  7,  0,  0, 12, 0},
{16, 6,   4,  0,  5,  2,  0,  0,  1,  0,  0,  0, 11,  0,  0, 12},
{5,  11,  0,  0,  0,  3,  0, 15,  0, 16,  0, 13,  0,  1,  0, 8},
{0,  0,   3,  0,  0,  6, 11, 14,  0,  5,  7,  0,  0,  9,  0, 0},
{0,  0,   0, 14,  8,  0, 10,  0,  0, 11, 12,  0,  0,  0,  0, 0},
{0,  7,  13,  0,  0,  0,  0, 12,  0,  8,  9,  0,  0,  0,  3, 0},
{0,  0,  11,  9,  0,  7,  0,  0,  0,  0,  0, 12,  0,  8, 16, 5},
{0,  0,  10,  0, 11, 13,  0,  0,  0,  0,  0,  3, 12,  0,  6, 0},
{0,  5,   0,  0, 10, 15,  0,  1,  7,  2,  0,  0, 14, 11,  0, 0},
{0,  0,   5,  0,  0, 12, 14,  0,  0, 10,  0,  0, 15,  0,  0, 4},
{9,  0,  14,  6,  0,  0,  1,  0, 16,  0,  2,  0,  3,  0, 13, 0},
{8,  13,  0,  4,  0,  0,  0,  0, 12,  7,  3,  0,  0,  6,  0, 0},
{0,  16, 12,  0,  0,  5,  0,  9,  0, 13, 14,  4,  1,  0,  0, 0} };
/**/
```
```   int grid[N][N] =
{ {1,  0,   4,  0, 25,  0, 19,  0,  0,  10,  21, 8,  0,  14, 0,  6,  12,   9,  0,  0,  0,  0,  0,  0,  5},{5,  0,  19, 23, 24,  0, 22,  12,  0,  0,  16, 6,  0,  20,  0,  18,  0,   25,  14,  13,  10, 11,  0,  1,  15},{0,  0,   0,  0,  0,  0,  21,  5,  0,  20,  11,  10,  0,  1,  0,  4,  8,   24,  23,  15,  18,  0,  16,  22,  19},
```

{0, 7, 21, 8, 18, 0, 0, 0, 11, 0, 5, 0, 0, 24, 0, 0, 0, 17, 22, 1, 9, 6, 25, 0, 0}, {0, 13, 15, 0, 22, 14, 0, 18, 0, 16, 0, 0, 0, 4, 0, 0, 0, 19, 0, 0, 0, 24, 20, 21, 17}, {12, 0, 11, 0, 6, 0, 0, 0, 0, 15, 0, 0, 0, 0, 21, 25, 19, 0, 4, 0, 22, 14, 0, 20, 0}, {8, 0, 0, 21, 0, 16, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 17, 23, 18, 22, 0, 0, 0, 24, 6}, {4, 0, 14, 18, 7, 9, 0, 22, 21, 19, 0, 0, 0, 2, 0, 5, 0, 0, 0, 6, 16, 15, 0, 11, 12}, {22, 0, 24, 0, 23, 0, 0, 11, 0, 7, 0, 0, 4, 0, 14, 0, 2, 12, 0, 8, 5, 19, 0, 25, 9}, {20, 0, 0, 0, 5, 0, 0, 0, 0, 17, 9, 0, 12, 18, 0, 1, 0, 0, 7, 24, 0, 0, 0, 13, 4}, {13, 0, 0, 5, 0, 2, 23, 14, 4, 18, 22, 0, 17, 0, 0, 20, 0, 1, 9, 21, 12, 0, 0, 8, 11}, {14, 23, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 20, 25, 0, 3, 4, 13, 0, 11, 21, 9, 5, 18, 22}, {7, 0, 0, 11, 17, 20, 24, 0, 0, 0, 3, 4, 1, 12, 0, 0, 6, 14, 0, 5, 25, 13, 0, 0, 0}, {0, 0, 16, 9, 0, 17, 11, 7, 10, 25, 0, 0, 0, 13, 6, 0, 0, 18, 0, 0, 19, 4, 0, 0, 20}, {6, 15, 0, 19, 4, 13, 0, 0, 5, 0, 18, 11, 0, 0, 9, 8, 22, 16, 25, 10, 7, 0, 0, 0, 0}, {0, 0, 0, 2, 0, 0, 10, 19, 3, 0, 1, 0, 22, 9, 4, 11, 15, 0, 20, 0, 0, 8, 23, 0, 25}, {0, 24, 8, 13, 1, 0, 0, 4, 20, 0, 17, 14, 0, 0, 18, 0, 16, 22, 5, 0, 11, 0, 10, 0, 0}, {23, 10, 0, 0, 0, 0, 0, 0, 18, 0, 6, 0, 16, 0, 0, 17, 1, 0, 13, 0, 0, 3, 19, 12, 0}, {25, 5, 0, 14, 11, 0, 17, 0, 8, 24, 13, 0, 19, 23, 15, 9, 0, 0, 12, 0, 20, 0, 22, 0, 7}, {0, 0, 17, 4, 0, 22, 15, 0, 23, 11, 12, 25, 0, 0, 0, 0, 18, 8, 0, 7, 0, 0, 14, 0, 13}, {19, 6, 23, 22, 8, 0, 0, 1, 25, 4, 14, 2, 0, 3, 7, 13, 10, 11, 16, 0, 0, 0, 0, 0, 0}, {0, 4, 0, 17, 0, 3, 0, 24, 0, 8, 20, 23, 11, 10, 25, 22, 0, 0, 0, 12, 13, 2, 18, 6, 0}, {0, 0, 7, 16, 0, 0, 6, 17, 2, 21, 0, 18, 0, 0, 0, 19, 0, 0, 8, 0, 0, 0, 0, 4, 0}, {18, 9, 25, 1, 2, 11, 0, 0, 13, 22, 4, 0, 21, 0, 5, 0, 23, 7, 0, 0, 15, 0, 3, 0, 8}, {0, 21, 10, 0, 0, 12, 0, 20, 16, 0, 19, 0, 0, 0, 0, 15, 14, 4, 2, 18, 23, 25, 11, 7, 0} }; /**/

```   int* d_a;      //Table
int* d_result; //Table change indicator
```
```   cudaMalloc((void**)&d_a, N*N * sizeof(int));
cudaMalloc((void**)&d_result, sizeof(int));
```
```   //Copy Sudoku over
cudaMemcpy(d_a, grid, N*N * sizeof(int), cudaMemcpyHostToDevice);
```
```   SolveSudoku(grid, d_a, d_result);
```
```   //Copy Sudoku back
cudaMemcpy(grid, d_a, N*N * sizeof(int), cudaMemcpyDeviceToHost);
printGrid(grid);
```
```   cudaFree(d_a);
cudaFree(d_result);
```
```   return 0;
```

}

Single Pass Sudoku Solver

```This Kernel was designed to run on a single block with dimensions N*N the size of the Sudoku
limiting us to a Sudoku of size 25 * 25
For each empty space, counts the number possible values which can fit and how many times each value can fit in that section
If only one value can fit or that value has only one place, assigns the value
```

```__global__ void superSolve(int * d_a) {

//Used to remember which row | col | box ( section ) have which values
__shared__ bool rowHas[N][N];
__shared__ bool colHas[N][N];
__shared__ bool boxHas[N][N];

//Used to ensure that the table has changed
__shared__ int added, past;

//Number of spaces which can place the number in each section
__shared__ int rowCount[N][N];
__shared__ int colCount[N][N];
__shared__ int boxCount[N][N];

//Where the square is located in the Sudoku
int row = threadIdx.x;
int col = threadIdx.y;
int box = row / BOXWIDTH + (col / BOXWIDTH) * BOXWIDTH;

//Unique identifier for each square in row, col, box
//Corresponds to the generic Sudoku Solve
//Using a Sudoku to solve a Sudoku !!!
int offset = col + (row % BOXWIDTH) * BOXWIDTH + (box % BOXWIDTH);

//Square's location in the Sudoku
int gridIdx = col * N + row;

int at = d_a[gridIdx];

if (!gridIdx) { //Thread at 0,0 sets values
added = -1;
past = -2;
}

rowHas[col][row] = false;
colHas[col][row] = false;
boxHas[col][row] = false;
__syncthreads();

if (at != UNASSIGNED) {
rowHas[row][at - 1] = true;
colHas[col][at - 1] = true;
boxHas[box][at - 1] = true;
}

//Previous loop has not changed any values
while (added != past) {
//RESET counters
rowCount[col][row] = 0;
colCount[col][row] = 0;
boxCount[col][row] = 0;
__syncthreads();

if (!gridIdx) //forget previous change
past = added;
int count = 0;  //number of values which can fit in this square
int guess = at; //last value found which can fit in this square
for (int idx = 0; idx < N; ++idx) {
//Ensures that every square in each section is working on a different number in the section
int num = (idx + offset) % N;
if (at == UNASSIGNED && !(rowHas[row][num] || colHas[col][num] || boxHas[box][num])) {
count++;
guess = num + 1;
rowCount[row][num] ++;
colCount[col][num] ++;
boxCount[box][num] ++;
}
__syncthreads();
}

//Only ONE value can fit in this spot
if (count == 1) {
at = guess--;
d_a[gridIdx] = at;
rowHas[row][guess] = true;
colHas[col][guess] = true;
boxHas[box][guess] = true;
added = gridIdx;
}
__syncthreads();

if (at == UNASSIGNED) {
//Find values which can go in only one spot in the section
for (int idx = 0; idx < N; ++idx) {
if (!(rowHas[row][idx] || colHas[col][idx] || boxHas[box][idx]) &&
(boxCount[box][idx] == 1 || rowCount[row][idx] == 1 || colCount[col][idx] == 1)) {
//In this section this value can only appear in this square
at = idx + 1;
d_a[gridIdx] = at;
rowHas[row][idx] = true;
colHas[col][idx] = true;
boxHas[box][idx] = true;
added = gridIdx;
}
}
}
__syncthreads();
}
}
```

Assignment 3

```Changes:
Reduced Thread Divergence/CGMA
-each thread now remembers which values it has seen in a boolean array
- values are only assigned to the grid after the kernel 'solves' the sudoku
- at value in kernel and shared memory for rowHas, colHas, boxHas, updated in a single place
Coalesced Memory
- change modifying _Has and _Count arrays from row->col to col->row as row(threadIdx.x) is our fastest moving dimension
Clarified Code
- use gridIdx == 0 rather then !gridIdx
- use a do-while loop rather then a while loop
```
Full code
```#include <stdio.h>
// CUDA header file
#include "cuda_runtime.h"
#include <device_launch_parameters.h>
#ifndef __CUDACC__
#define __CUDACC__
#endif
// UNASSIGNED is used for empty cells in Sudoku grid
#define UNASSIGNED 0

// BOX_W is used for the length of one of the square sub-regions of the Sudoku grid.
// Overall length will be N * N.
#define BOX_W 5
#define N (BOX_W * BOX_W)

__global__ void solve(int* d_a) {
// Used to remember which row | col | box ( section ) have which values
__shared__ bool rowHas[N][N];
__shared__ bool colHas[N][N];
__shared__ bool boxHas[N][N];

// Used to ensure that the table has changed
__shared__ bool changed;

// Number of spaces which can place the number in each section
__shared__ int rowCount[N][N];
__shared__ int colCount[N][N];
__shared__ int boxCount[N][N];

// Where the square is located in the Sudoku
int row = threadIdx.x;
int col = threadIdx.y;
int box = row / BOX_W + (col / BOX_W) * BOX_W;

// Unique identifier for each square in row, col, box
// Corresponds to the generic Sudoku Solve
// Using a Sudoku to solve a Sudoku !!!
int offset = col + (row % BOX_W) * BOX_W + (box % BOX_W);

// Square's location in the Sudoku
int gridIdx = col * N + row;

int at = d_a[gridIdx];

bool notSeen[N];
for (int i = 0; i < N; ++i)
notSeen[i] = true;

rowHas[col][row] = false;
colHas[col][row] = false;
boxHas[col][row] = false;
__syncthreads();

if (at != UNASSIGNED) {
rowHas[row][at - 1] = true;
colHas[col][at - 1] = true;
boxHas[box][at - 1] = true;
}
// Previous loop has not changed any values
do {
// RESET counters
rowCount[col][row] = 0;
colCount[col][row] = 0;
boxCount[col][row] = 0;
__syncthreads();

if (gridIdx == 0) // forget previous change
changed = false;
int count = 0;  // number of values which can fit in this square
int guess = 0; // last value found which can fit in this square
for (int idx = 0; idx < N; ++idx) {
// Ensures that every square in each section is working on a different number in the section
int num = (idx + offset) % N;
if (at == UNASSIGNED && notSeen[num]) {
if (rowHas[row][num] || boxHas[box][num] || colHas[col][num])
notSeen[num] = false;
else {
++count;
guess = num;
rowCount[row][num]++;
colCount[col][num]++;
boxCount[box][num]++;
}
}
__syncthreads();
}

// Find values which can go in only one spot in the section
for (int idx = 0; idx < N && count > 1; ++idx) {
if (notSeen[idx] &&
(rowCount[row][idx] == 1 || boxCount[box][idx] == 1 || colCount[col][idx] == 1)) {
// In this section this value can only appear in this square
guess = idx;
count = 1;
}
}

if (count == 1) {
at = guess + 1;
rowHas[row][guess] = true;
colHas[col][guess] = true;
boxHas[box][guess] = true;
changed = true;
}
__syncthreads();
} while (changed);

//SOLVED CHECK
if (!(rowHas[row][col] || colHas[row][col] || boxHas[row][col]))
changed = true;
__syncthreads();
if (changed && gridIdx == 0)
at = 0;

d_a[gridIdx] = at;
}

void print(int result[N][N]) {
for (int row = 0; row < N; row++) {
for (int col = 0; col < N; col++)
printf("%3d", result[row][col]);
printf("\n");
}
}

// Driver program to test main program functions
int main() {
int h_a[N][N] = {
{  1,  0,  4,  0, 25,  0, 19,  0,  0, 10, 21,  8,  0, 14,  0,  6, 12,  9,  0,  0,  0,  0,  0,  0,  5},
{  5,  0, 19, 23, 24,  0, 22, 12,  0,  0, 16,  6,  0, 20,  0, 18,  0, 25, 14, 13, 10, 11,  0,  1, 15},
{  0,  0,  0,  0,  0,  0, 21,  5,  0, 20, 11, 10,  0,  1,  0,  4,  8, 24, 23, 15, 18,  0, 16, 22, 19},
{  0,  7, 21,  8, 18,  0,  0,  0, 11,  0,  5,  0,  0, 24,  0,  0,  0, 17, 22,  1,  9,  6, 25,  0,  0},
{  0, 13, 15,  0, 22, 14,  0, 18,  0, 16,  0,  0,  0,  4,  0,  0,  0, 19,  0,  0,  0, 24, 20, 21, 17},
{ 12,  0, 11,  0,  6,  0,  0,  0,  0, 15,  0,  0,  0,  0, 21, 25, 19,  0,  4,  0, 22, 14,  0, 20,  0},
{  8,  0,  0, 21,  0, 16,  0,  0,  0,  2,  0,  3,  0,  0,  0,  0, 17, 23, 18, 22,  0,  0,  0, 24,  6},
{  4,  0, 14, 18,  7,  9,  0, 22, 21, 19,  0,  0,  0,  2,  0,  5,  0,  0,  0,  6, 16, 15,  0, 11, 12},
{ 22,  0, 24,  0, 23,  0,  0, 11,  0,  7,  0,  0,  4,  0, 14,  0,  2, 12,  0,  8,  5, 19,  0, 25,  9},
{ 20,  0,  0,  0,  5,  0,  0,  0,  0, 17,  9,  0, 12, 18,  0,  1,  0,  0,  7, 24,  0,  0,  0, 13,  4},
{ 13,  0,  0,  5,  0,  2, 23, 14,  4, 18, 22,  0, 17,  0,  0, 20,  0,  1,  9, 21, 12,  0,  0,  8, 11},
{ 14, 23,  0, 24,  0,  0,  0,  0,  0,  0,  0,  0, 20, 25,  0,  3,  4, 13,  0, 11, 21,  9,  5, 18, 22},
{  7,  0,  0, 11, 17, 20, 24,  0,  0,  0,  3,  4,  1, 12,  0,  0,  6, 14,  0,  5, 25, 13,  0,  0,  0},
{  0,  0, 16,  9,  0, 17, 11,  7, 10, 25,  0,  0,  0, 13,  6,  0,  0, 18,  0,  0, 19,  4,  0,  0, 20},
{  6, 15,  0, 19,  4, 13,  0,  0,  5,  0, 18, 11,  0,  0,  9,  8, 22, 16, 25, 10,  7,  0,  0,  0,  0},
{  0,  0,  0,  2,  0,  0, 10, 19,  3,  0,  1,  0, 22,  9,  4, 11, 15,  0, 20,  0,  0,  8, 23,  0, 25},
{  0, 24,  8, 13,  1,  0,  0,  4, 20,  0, 17, 14,  0,  0, 18,  0, 16, 22,  5,  0, 11,  0, 10,  0,  0},
{ 23, 10,  0,  0,  0,  0,  0,  0, 18,  0,  6,  0, 16,  0,  0, 17,  1,  0, 13,  0,  0,  3, 19, 12,  0},
{ 25,  5,  0, 14, 11,  0, 17,  0,  8, 24, 13,  0, 19, 23, 15,  9,  0,  0, 12,  0, 20,  0, 22,  0,  7},
{  0,  0, 17,  4,  0, 22, 15,  0, 23, 11, 12, 25,  0,  0,  0,  0, 18,  8,  0,  7,  0,  0, 14,  0, 13},
{ 19,  6, 23, 22,  8,  0,  0,  1, 25,  4, 14,  2,  0,  3,  7, 13, 10, 11, 16,  0,  0,  0,  0,  0,  0},
{  0,  4,  0, 17,  0,  3,  0, 24,  0,  8, 20, 23, 11, 10, 25, 22,  0,  0,  0, 12, 13,  2, 18,  6,  0},
{  0,  0,  7, 16,  0,  0,  6, 17,  2, 21,  0, 18,  0,  0,  0, 19,  0,  0,  8,  0,  0,  0,  0,  4,  0},
{ 18,  9, 25,  1,  2, 11,  0,  0, 13, 22,  4,  0, 21,  0,  5,  0, 23,  7,  0,  0, 15,  0,  3,  0,  8},
{  0, 21, 10,  0,  0, 12,  0, 20, 16,  0, 19,  0,  0,  0,  0, 15, 14,  4,  2, 18, 23, 25, 11,  7,  0}
};

int* d_a;      //Table
cudaMalloc((void**)&d_a, N * N * sizeof(int));
// Copy Sudoku to device
cudaMemcpy(d_a, h_a, N * N * sizeof(int), cudaMemcpyHostToDevice);
dim3 dBlock(N, N);
solve << <1, dBlock >> > (d_a);
// Copy Sudoku back to host
cudaMemcpy(h_a, d_a, N * N * sizeof(int), cudaMemcpyDeviceToHost);
// Check if solved
if (h_a[0][0])
print(h_a);
else
printf("No solution could be found.");
cudaFree(d_a);
return 0;
}
```

Kernel Optimization Attempts

These Kernels change a minor part of the Optimized Kernel or use a slightly different algorithm in an attempt to make it faster

```Change : Replaces the boolean array hasSeen with a single int & uses bitwise operators
Theory : Since local array variables of threads are stored in Global memory this was an attempt to move that into a register
Result : No speed up noticed, suggesting that more is happening beyond arrays stored in Global memory, perhaps some type of paging,
more testing would be needed on something less erratic then a Sudoku Solver
```
Using a int as a boolean array (Kernel)
```__global__ void solve(int* d_a) {
// Used to remember which row | col | box ( section ) have which values
__shared__ bool rowHas[N][N];
__shared__ bool colHas[N][N];
__shared__ bool boxHas[N][N];
// Used to ensure that the table has changed
__shared__ bool changed;
// Number of spaces which can place the number in each section
__shared__ int rowCount[N][N];
__shared__ int colCount[N][N];
__shared__ int boxCount[N][N];
// Where the square is located in the Sudoku
int row = threadIdx.x;
int col = threadIdx.y;
int box = row / BOX_W + (col / BOX_W) * BOX_W;

int gridIdx = col * N + row;
int at = d_a[gridIdx];
// Unique identifier for each square in row, col, box
// Corresponds to the generic Sudoku Solve
// Using a Sudoku to solve a Sudoku !!!
int offset = col + (row % BOX_W) * BOX_W + (box % BOX_W);
// Square's location in the Sudoku

int notSeen = 0;
rowHas[col][row] = false;
colHas[col][row] = false;
boxHas[col][row] = false;
__syncthreads();
if (at != UNASSIGNED) {
rowHas[row][at - 1] = true;
colHas[col][at - 1] = true;
boxHas[box][at - 1] = true;
} else {
notSeen = ~0;
}
__syncthreads();

// Previous loop has not changed any values
do {
// RESET counters
rowCount[col][row] = 0;
colCount[col][row] = 0;
boxCount[col][row] = 0;
__syncthreads();
if (gridIdx == 0) // forget previous change
changed = false;
int count = 0;  // number of values which can fit in this square
int guess = 0; // last value found which can fit in this square
int b_shuttle = 1;
for (int idx = 0; idx < N; ++idx) {
// Ensures that every square in each section is working on a different number in the section
int num = (idx + offset) % N;
if (b_shuttle & notSeen) {
if (rowHas[row][num] || boxHas[box][num] || colHas[col][num])
notSeen ^= b_shuttle;
else {
++count;
guess = num;
rowCount[row][num]++;
colCount[col][num]++;
boxCount[box][num]++;
}
}
b_shuttle <<= 1;
__syncthreads();
}
// Find values which can go in only one spot in the section

b_shuttle = 1;
for (int idx = 0; idx < N && count > 1; ++idx) {
int num = (idx + offset) % N;
if ((b_shuttle & notSeen) &&
(rowCount[row][num] == 1 || boxCount[box][num] == 1 || colCount[col][num] == 1)) {
// In this section this value can only appear in this square
guess = num;
count = 1;
}
b_shuttle <<= 1;
}
```
```         if (count == 1) {
at = guess + 1;
notSeen = 0;
rowHas[row][guess] = true;
colHas[col][guess] = true;
boxHas[box][guess] = true;
changed = true;
}
__syncthreads();
} while (changed);
//SOLVED CHECK
if (!(rowHas[row][col] || colHas[row][col] || boxHas[row][col]))
changed = true;
__syncthreads();
if (changed && gridIdx == 0)
at = 0;

d_a[gridIdx] = at;
}
```
```Change : Remove the counters, and logic which checks for a section needing a value in one place
Theory : The counting logic requires a additional nested loop each solve cycle and created more thread divergence
Result : The algorithm is slower, probably because 'sections requiring a single value' adds more values early in the kernel resulting in less passes overall
Also this kernel is similar to one of my earlier builds, which was unable to solve the 9x9 getting stuck on every square having more then one possible value
```
Dropping Section Logic (Kernel)
```__global__ void solve(int* d_a) {
// Used to remember which row | col | box ( section ) have which values
__shared__ bool rowHas[N][N];
__shared__ bool colHas[N][N];
__shared__ bool boxHas[N][N];

// Used to ensure that the table has changed
__shared__ bool changed;

// Number of spaces which can place the number in each section

// Where the square is located in the Sudoku
int row = threadIdx.x;
int col = threadIdx.y;
int box = row / BOX_W + (col / BOX_W) * BOX_W;

// Unique identifier for each square in row, col, box
// Corresponds to the generic Sudoku Solve
// Using a Sudoku to solve a Sudoku !!!
int offset = col + (row % BOX_W) * BOX_W + (box % BOX_W);

// Square's location in the Sudoku
int gridIdx = col * N + row;

int at = d_a[gridIdx];

bool notSeen[N];
for (int i = 0; i < N; ++i)
notSeen[i] = true;

rowHas[col][row] = false;
colHas[col][row] = false;
boxHas[col][row] = false;
__syncthreads();

if (at != UNASSIGNED) {
rowHas[row][at - 1] = true;
colHas[col][at - 1] = true;
boxHas[box][at - 1] = true;
}
// Previous loop has not changed any values
do {
// RESET counters
__syncthreads();

if (gridIdx == 0) // forget previous change
changed = false;
int count = 0;  // number of values which can fit in this square
int guess = 0; // last value found which can fit in this square
for (int idx = 0; idx < N; ++idx) {
// Ensures that every square in each section is working on a different number in the section
int num = (idx + offset) % N;
if (at == UNASSIGNED && notSeen[num]) {
if (rowHas[row][num] || boxHas[box][num] || colHas[col][num])
notSeen[num] = false;
else {
++count;
guess = num;
}
}
__syncthreads();
}

if (count == 1) {
at = guess + 1;
rowHas[row][guess] = true;
colHas[col][guess] = true;
boxHas[box][guess] = true;
changed = true;
}
__syncthreads();
} while (changed);

//SOLVED CHECK
if (!(rowHas[row][col] || colHas[row][col] || boxHas[row][col]))
changed = true;
__syncthreads();
if (changed && gridIdx == 0)
at = 0;

d_a[gridIdx] = at;
}
```
```Change : Quickly finds one section that requires a single value in one spot, by checking all sections at once and remembering a single value
Theory : Similar to the previous Kernel, trying to remove the second loop
Result : Surprisingly slow, gains little benefit from the section logic
```
Notify (Kernel) Determines a single section that has a limited value (removes section loop)
```__global__ void solve(int* d_a) {
// Used to remember which row | col | box ( section ) have which values
__shared__ bool rowHas[N][N];
__shared__ bool colHas[N][N];
__shared__ bool boxHas[N][N];

// Used to ensure that the table has changed
__shared__ bool changed;

// Number of spaces which can place the number in each section
__shared__ int rowCount[N][N];
__shared__ int colCount[N][N];
__shared__ int boxCount[N][N];

// Where the square is located in the Sudoku
int row = threadIdx.x;
int col = threadIdx.y;
int box = row / BOX_W + (col / BOX_W) * BOX_W;

// Unique identifier for each square in row, col, box
// Corresponds to the generic Sudoku Solve
// Using a Sudoku to solve a Sudoku !!!
int offset = col + (row % BOX_W) * BOX_W + (box % BOX_W);

// Square's location in the Sudoku
int gridIdx = col * N + row;

int at = d_a[gridIdx];

bool notSeen[N];
for (int i = 0; i < N; ++i)
notSeen[i] = true;

rowHas[col][row] = false;
colHas[col][row] = false;
boxHas[col][row] = false;
__syncthreads();
__shared__ int notify;
if (at != UNASSIGNED) {
rowHas[row][at - 1] = true;
colHas[col][at - 1] = true;
boxHas[box][at - 1] = true;
}
// Previous loop has not changed any values
do {
// RESET counters
rowCount[col][row] = 0;
colCount[col][row] = 0;
boxCount[col][row] = 0;
__syncthreads();

if (gridIdx == 0) { // forget previous change
changed = false;
notify = -1;
}
int count = 0;  // number of values which can fit in this square
int guess = 0; // last value found which can fit in this square
for (int idx = 0; idx < N; ++idx) {
// Ensures that every square in each section is working on a different number in the section
int num = (idx + offset) % N;
if (at == UNASSIGNED && notSeen[num]) {
if (rowHas[row][num] || boxHas[box][num] || colHas[col][num])
notSeen[num] = false;
else {
++count;
guess = num;
rowCount[row][num]++;
colCount[col][num]++;
boxCount[box][num]++;
}
}
__syncthreads();
}
if (rowCount[row][col] == 1 || colCount[row][col] == 1 || boxCount[row][col] == 1)
notify = col;
__syncthreads();
// Find values which can go in only one spot in the section

if (notify > 0 && at == UNASSIGNED && notSeen[notify] &&
(rowCount[row][notify] == 1 || boxCount[box][notify] == 1 || colCount[col][notify] == 1)) {
// In this section this value can only appear in this square
guess = notify;
count = 1;
}

if (count == 1) {
at = guess + 1;
rowHas[row][guess] = true;
colHas[col][guess] = true;
boxHas[box][guess] = true;
changed = true;
}
__syncthreads();
} while (changed);

//SOLVED CHECK
if (!(rowHas[row][col] || colHas[row][col] || boxHas[row][col]))
changed = true;
__syncthreads();
if (changed && gridIdx == 0)
at = 0;

d_a[gridIdx] = at;
}
```
CountDown - Counts down instead of revisiting numbers
```__global__ void solve(int* d_a) {
// Used to remember which row | col | box ( section ) have which values
__shared__ bool rowHas[N][N];
__shared__ bool colHas[N][N];
__shared__ bool boxHas[N][N];
// Used to ensure that the table has changed
__shared__ bool changed;

// Number of spaces which can place the number in each section
__shared__ int rowCount[N][N];
__shared__ int colCount[N][N];
__shared__ int boxCount[N][N];
// Where the square is located in the Sudoku
int row = threadIdx.x;
int col = threadIdx.y;
int box = row / BOX_W + (col / BOX_W) * BOX_W;

int gridIdx = col * N + row;
int at = d_a[gridIdx];

// Unique identifier for each square in row, col, box
// Corresponds to the generic Sudoku Solve
// Using a Sudoku to solve a Sudoku !!!
int offset = col + (row % BOX_W) * BOX_W + (box % BOX_W);
// Square's location in the Sudoku

int count = 0; //Number of values which can fit in this square
int notSeen = 0; //Boolean Array as an Integer

if (gridIdx == 0)
changed = true;
```
```   rowHas[col][row] = false;
colHas[col][row] = false;
boxHas[col][row] = false;

rowCount[col][row] = 0;
colCount[col][row] = 0;
boxCount[col][row] = 0;

__syncthreads();
if (at != UNASSIGNED) {
rowHas[row][at - 1] = true;
colHas[col][at - 1] = true;
boxHas[box][at - 1] = true;
}
__syncthreads();
int guess;
int b_shuttle = 1;
for (int idx = 0; idx < N; ++idx) {
int num = (idx + offset) % N;
if (at == UNASSIGNED && !(rowHas[row][num] || boxHas[box][num] || colHas[col][num])) {
notSeen |= b_shuttle;    //this value can go here
++count;                 //how many values this square can have
guess = num;
//how many values this section can have
rowCount[row][num]++;
colCount[col][num]++;
boxCount[box][num]++;
}
__syncthreads();
b_shuttle <<= 1;
}

if (at == UNASSIGNED && count == 0) //NOT POSSIBLE SUDOKU
changed = false;
__syncthreads();

if (count == 1) {
at = guess + 1;
notSeen = count = 0;
rowHas[row][guess] = true;
colHas[col][guess] = true;
boxHas[box][guess] = true;
}
// Previous loop has not changed any values

while (changed) {
__syncthreads();
if (gridIdx == 0) // forget previous change
changed = false;
bool inSection = true;
int b_shuttle = 1;
for (int idx = 0; idx < N; ++idx) {
// Ensures that every square in each section is working on a different number in the section
int num = (idx + offset) % N;
if (b_shuttle & notSeen) {
if (rowHas[row][num] || boxHas[box][num] || colHas[col][num]) {
notSeen ^= b_shuttle;
--count;
rowCount[row][num]--;
colCount[col][num]--;
boxCount[box][num]--;
}
else if (inSection) {
guess = num;
}
}
__syncthreads();
if ((b_shuttle & notSeen) && (rowCount[row][num] == 1 || boxCount[box][num] == 1 || colCount[col][num] == 1))
inSection = false;

b_shuttle <<= 1;
}

if (count == 1 || !inSection) {
at = guess + 1;
notSeen = count = 0;
rowHas[row][guess] = true;
colHas[col][guess] = true;
boxHas[box][guess] = true;
changed = true;
}
__syncthreads();
};

if (!(rowHas[row][col] && colHas[row][col] && boxHas[box][col]))
changed = true; //HAVE NOT SOLVED the sudoku

__syncthreads();
if (changed && gridIdx == 0)
at = 0;
d_a[gridIdx] = at;
}
```
Countdown Boolean Array
```__global__ void solve(int* d_a) {
// Used to remember which row | col | box ( section ) have which values
__shared__ bool rowHas[N][N];
__shared__ bool colHas[N][N];
__shared__ bool boxHas[N][N];
// Used to ensure that the table has changed
__shared__ bool changed;

// Number of spaces which can place the number in each section
__shared__ int rowCount[N][N];
__shared__ int colCount[N][N];
__shared__ int boxCount[N][N];
// Where the square is located in the Sudoku
int row = threadIdx.x;
int col = threadIdx.y;
int box = row / BOX_W + (col / BOX_W) * BOX_W;

int gridIdx = col * N + row;
int at = d_a[gridIdx];

// Unique identifier for each square in row, col, box
// Corresponds to the generic Sudoku Solve
// Using a Sudoku to solve a Sudoku !!!
int offset = col + (row % BOX_W) * BOX_W + (box % BOX_W);
// Square's location in the Sudoku

int count = 0; //Number of values which can fit in this square
bool notSeen[N]; //Boolean Array as an Integer
for (int i = 0; i < N; ++i)
notSeen[i] = false;

if (gridIdx == 0)
changed = true;
```
```   rowHas[col][row] = false;
colHas[col][row] = false;
boxHas[col][row] = false;

rowCount[col][row] = 0;
colCount[col][row] = 0;
boxCount[col][row] = 0;

__syncthreads();
if (at != UNASSIGNED) {
rowHas[row][at - 1] = true;
colHas[col][at - 1] = true;
boxHas[box][at - 1] = true;
}
__syncthreads();
int guess;
for (int idx = 0; idx < N; ++idx) {
int num = (idx + offset) % N;
if (at == UNASSIGNED && !(rowHas[row][num] || boxHas[box][num] || colHas[col][num])) {
notSeen[num] = true;    //this value can go here
++count;                //how many values this square can have
guess = num;
//how many values this section can have
rowCount[row][num]++;
colCount[col][num]++;
boxCount[box][num]++;
}
__syncthreads();
}

if (at == UNASSIGNED && count == 0) //NOT POSSIBLE SUDOKU
changed = false;
__syncthreads();

if (count == 1) {
at = guess + 1;
count = 0;
rowHas[row][guess] = true;
colHas[col][guess] = true;
boxHas[box][guess] = true;
}
// Previous loop has not changed any values

while (changed) {
__syncthreads();
if (gridIdx == 0) // forget previous change
changed = false;
bool inSection = true;
for (int idx = 0; idx < N; ++idx) {
// Ensures that every square in each section is working on a different number in the section
int num = (idx + offset) % N;
if (at == UNASSIGNED && notSeen[num]) {
if (rowHas[row][num] || boxHas[box][num] || colHas[col][num]) {
notSeen[num] = false;
--count;
rowCount[row][num]--;
colCount[col][num]--;
boxCount[box][num]--;
}
else if (inSection) {
guess = num;
}
}
__syncthreads();
if (at == UNASSIGNED && notSeen[num] && (rowCount[row][num] == 1 || boxCount[box][num] == 1 || colCount[col][num] == 1))
inSection = false;

}

if (count == 1 || !inSection) {
at = guess + 1;
count = 0;
rowHas[row][guess] = true;
colHas[col][guess] = true;
boxHas[box][guess] = true;
changed = true;
}
__syncthreads();
};

if (!(rowHas[row][col] && colHas[row][col] && boxHas[box][col]))
changed = true; //HAVE NOT SOLVED the sudoku
__syncthreads();
if (changed && gridIdx == 0)
at = 0;
d_a[gridIdx] = at;
}
```

For 9x9:

For 16x16:

For 25x25: